DUBLIN UNIVERSITY PRESS SERIES.
A HISTORY
OF THE
THEORIES OF AETHER AND ELECTRICITY
FKOM THE AGE OF DESCAKTES TO THE CLOSE OF
THE NINETEENTH CENTURY.
BY
E. T. WH1TTAKER,
Hon. Sc.D. (DubL}; I.E.S.; Roy at Astronomer of Ireland.
LONGMANS, GREEN, AND CO.,
39 PATERNOSTER ROW, LONDON,
NEW YORK, BOMBAY, AND CALCUTTA.
HODGES, FIGGIS, & CO., LTD., DUBLIN.
1910.
MM*
DUBLIN :
PRINTED AT UHE UNIVERSITY PRESS,
BY PONSONBY AND OIBRS.
THE author desires to record his gratitude to Mr. W. W.
EOUSE BALL, Fellow of Trinity College, Cambridge, and to
Professor W. McF. ORR, F.R.S., of the Royal College of Science
for Ireland ; these friends have read the proof-sheets, and have
made many helpful suggestions and criticisms.
Thanks are also 'due to the BOARD OF TRINITY COLLEGE,
DUBLIN, for the financial assistance which made possible the
publication of the work.
236360
CONTENTS.
CHAPTEK I.
y THE THEORY OF THE AETHER IN THE SEVENTEENTH CENTURY.
Page
Matter and aether, . . . . . . .1
The physical writings of Descartes, ..... 2
Early history of magnetism : Petrus Peregrinus, Gilbert, Descartes, 7
Fermat attacks Descartes' theory of light : the principle of least
time, ........ 10
Hooke's undulat>ry theory : the advance of wave -fronts, . . 11
Newton overthrows Hooke's theory of colours, . . .15
Conception of the aether in the writings of Newton, . . 17
Newton's theories of the periodicity of homogeneous light, and of
fits of easy transmission, • • . . ,20
The velocity of light : Galileo, Roemer, . . . .21
Huygens' Traite de la lumiere : his theories of the propagation of
waves, and of crystalline optics, • . . .22
Newton shows that rays obtained by double refraction have sides :
his objections to the undulatory theory, . . .28
X
CHAPTER II.
ELECTRIC AND MAGNETIC SCIENCE, PRIOR TO THE INTRODUCTION OF
THE POTENTIALS.
The electrical researches of Gilbert : the theory of emanations, . 29
State of physical science in the first half of the eighteenth century, 32
Gray discovers electric conduction : Desaguliers, . . • 37
The electric fluid, ....... 38
Du Fay distinguishes vitreous and resinous electricity, . .39
Xollet's effluent and affluent streams, . . . .40
The Leyden phial, ..... . . 41
The one-fluid theory : ideas of Watson and Franklin, . . 42
Final overthrow by Aepinus of the doctrine of effluvia, . . 48
Priestley discovers the law of electrostatic force, . . .50
viii Contents.
Page
Cavendish, . ... 51
Michell discovers the law of magnetic force, . . . .54
The two-fluid theory : Coulomb, . . . . .56
Limited mobility of the magnetic fluids, . . .58
Poisson's mathematical theory of electrostatics, . . .59
The equivalent surface- and volume-distributions of magnetism :
Poisson's theory of magnetic induction, . . .64
Green's Nottingham memoir, . . . . .65
CHAPTER III.
GALVANISM, FROM GALVANI TO OHM.
Sulzer's discovery, ... . .67
Galvanic phenomena, ....... 68
Rival hypotheses regarding the galvanic fluid, , . .70
The voltaic pile, ....... 72
Nicholson and Carlisle decompose water voltaically, . . 75
Davy's chemical theory of the pile, ..... 76
Grothuss' chain, . . . . . . .78
De La Rive's hypothesis, . . . . . .79
Berzelius' scheme of electro-chemistry, . . . .80
Early attempts to discover a connexion between electricity and
magnetism, . ... 83
Oersted's experiment : his explanation of it, . . .85
The law of Biot and Savart, . . . . . .86
The researches of Ampere on electrodynamics, . . 87
Seebeck's phenomenon, . . . . . .90
Davy's researches on conducting power, . . . .94
Ohm's theory : electroscopic force, . . . . .95
CHAPTER IV.
THE LUMINIFEBOUS MEDIUM, FROM BRADLEY TO FRESNEL.
Bradley discovers aberration, . . . . .99
John Bernoulli's model of the aether, .... 100
Maupertuis and the principle of least action, . . . 102
Views of Euler, Courtivron, Melvill, .... 104
Young defends the undulatory theory, and explains the colours of
thin plates, ... ... 105
Laplace supplies a corpuscular theory of double refraction, . . 109
Contents. ix
Page
Young proposes a dynamical theory of light in crystals, . . 110
Researches of Malus on polarization, .... Ill
Recognition of biaxal crystals, ... . 113
Fresnel successfully explains diffraction, . . . 114
His theory of the relative motion of aether and matter, . . 115
Young suggests the transversality of the vibrations of light, . 121
Fresnel discusses the dynamics of transverse vibrations, . . 123
Fresnel's theory of the propagation of light in crystals, . • . 125
Hamilton predicts conical refraction, . . . ] 31
Fresnel's theory of reflexion, ..... 133
CHAPTER V.
I,THE AETHER AS AN ELASTIC SOLID.
Astronomical objection to the elastic-solid theory : Stokes'
hypothesis. . . . . . . .137
Navier and Cauchy discover the equation of vibration of an elastic
solid, 139
Poisson distinguishes condensational and distortional waves, . 141
Cauchy's first and second theories of light iq, crystals, . . 143
Cauchy's first theory of reflexion, ..... 145
His second theory of reflexion, ..... 147
The theory of reflexion of MacCullagh and Neumann, . . 148
Green discovers the correct conditions at the boundaries, . . 151
Green's theory of reflexion : objections to it, . . . 152
MacCullagh introduces a new type of elastic solid, . . . 154
W. Thomson's model of a rotationally-elastic body, . . 157
Cauchy's third theory of reflexion : the contractile aether, . . 158
Later work of W. Thomson and others on the contractile aether, . 159
Green's first and second theories of light in crystals, . . 161
Influence of Green, ....... 167
Researches of Stokes on the relation of the direction of vibration of
light to its plane of polarization, .... 168
The hypothesis of aeolotropic inertia, .... 171
Rotation of the plane of polarization of light by active bodies, . 173
MacCullagh's theory of natural rotatory power, . . 175
MacCullagh's and Cauchy's theory of metallic reflexion, . . 177
Extension of the elastic -solid theory to metals, . . 179
Lord Rayleigh's objection, .... . 181
Cauchy's theory of dispersion, • . . 182
Boussinesq's elastic-solid theory, ..... 185
x Contents.
CHAPTEE VI.
FARADAY.
Page
Discovery of induced currents : lines of magnetic force, . . 189
Self-induction, . . . . . . .193
Identity of frictional and voltaic electricity : Faraday's views on the
nature of electricity, . . . . . 194
Electro-chemistry, . . ".. • . • *. . . . 197
Controversy between the adherents of the chemical and contact
hypotheses, . » . . . . . 201
The properties of dielectrics, . . . . . 206
Theory of dielectric polarization : Faraday, W. Thomson, and
Mossotti, . . : . . . . .211
The connexion between magnetism and light, . . . 213
Airy's theory of magnetic rotatory polarization, . « . 214
Faraday's Thoughts on Ray -Vibrations, . ..''-. . . 217
Researches of Faraday and Pliicker on diamagnetism, . . 218
CHAPTER VII.
THE MATHEMATICAL ELECTRICIANS OF THE MIDDLE OF THE NINETEENTH
CENTURY.
F. Neumann's theory of induced currents : the electrodynamic
potential, . . . . . ; . . 222
W. Weber's theory of electrons, . . . . .225
Riemann's law, . . . ... . 231
v-Proposals to modify the law of gravitation, . .. . . 232
Weber's theory of paramagnetism and diamagnetism : later theories, 234
Joule's law : energetics of the voltaic cell, .... 239
Researches of Helmholtz on electrostatic and electrodynamic energy, 242
W. Thomson distinguishes the circuital and irrotational magnetic
vectors, ........ 244
His theory of magnecrystallic action, .... 245
His formula for the energy of a magnetic field, . . . 247
Extension of this formula to the case of fields produced by currents, 249
Kirchhoff identifies Ohm's electroscopic force with electrostatic
potential, . . . . . . / 251
The discharge of a Leyden jar : W. Thomson's theory, . . 253
The velocity of electricity and the propagation of telegraphic signals, 254
Clausius' law of force between electric charges : crucial experiments, 261
Nature of the current, ...... 263
The thermo-electric researches of Peltier and W. Thomson, • 264
Contents. xi
CHAPTER VIII.
MAXWELL.
Page
Gauss and Riemann on the propagation of electric actions, . . 268
Analogies suggested by W. Thomson, .... 269
Maxwell's hydrodynamical analogy, ..... 271
The vector potential, ...... 273
Linear and rotatory interpretations of magnetism, . . . 274
Maxwell's mechanical model of the electromagnetic field, . . 276
Electric displacement, ...... 279
Similarity of electric vibrations to those of light, . . . 281
Connexion of refractive index and specific inductive capacity, . 283
Maxwell's memoir of 1864, . . ... .284
The propagation of electric disturbances in crystals and in metals, . 288
Anomalous dispersion, ...... 291
The Max well -Sellmeier theory of dispersion, . . . 292
Imperfections of the electromagnetic theory of light, . . 295
The theory of L. Lorenz, ...... 297
Maxwell's theory of stress in the electric field, . . . 300
The pressure of radiation, ...... 303
Maxwell's theory of the magnetic rotation of light, . . . 307
CHAPTER IX.
MODELS OF THE AETHER.
Analogies in which a rotatory character is attributed to magnetism, 310
Models in which magnetic force is represented as a linear velocity, 311
Researches of W. Thomson, Bjerknes, and Leahy, on pulsating and
oscillating bodies, ...... 316
MacCullagh's quasi-elastic solid as a model of the electric medium, 318
The Hall effect, . . . . . .320
Models of Riemann and Fitz Gerald, . . . . 324
Vortex-atoms, . . . . . . .326
The vortex-sponge theory of the aether : researches of W. Thomson,
Fitz Gerald, and Hicks, , . . . . .327
CHAPTER X.
THE FOLLOWERS OF MAXWELL.
Helmholtz and H. A. Lorentz supply an electromagnetic theory of
reflexion, ....... 337
Crucial experiments of Helmholtz and Schiller, . . . 338
xii Contents.
Page
Convection -currents : Rowland's experiments, . . . 339
The moving charged sphere : researches of J. J. Thomson, Fitz Gerald,
and Heaviside, . . . . . . . 340
Conduction of rapidly -alternating currents, .... 344
Fitz Gerald devises the magnetic radiator, .... 345
Poynting's theorem, ....... 347
Poynting and J. J. Thomson develop the theory of moving lines of
force, . . . . . . . 349
Mechanical momentum in the electromagnetic field, . . 352
New derivation of Maxwell's equations by Hertz, . . . 353
Hertz's assumptions and Weber's theory, .... 356
Experiments of Hertz on electric waves, .... 357
The memoirs of Hertz and Heaviside on fields in which material
bodies are in motion, ...... 365
The current of dielectric convection, ..... 367
Kerr's magneto-optic phenomenon, . ... 368
Rowland's theory of magneto-optics, .... 369
The rotation of the plane of polarization in naturally active bodies, 370
CHAPTER XI.
CONDUCTION IN SOLUTIONS AND GASES, FROM FARADAY TO
J. J. THOMSON.
The hypothesis of Williamson and Clausius, . . . 372
Migration of the ions, ...... 373
The researches of Hittorf and Kohlrausch, .... 374
Polarization of electrodes, ...... 375
Electrocapillarity, .... . 376
Single differences of potential, . . . . . 379
Helmholtz' theory of concentration-cells, .... 381
Arrhenius' hypothesis, ... ... 383
The researches of Nernst, ... . 386
Earlier investigations of the discharge in rarefied gases, . . 390
Faraday observes the dark space, ..... 391
Researches of Pliicker, Hittorf, Goldstein, and Varley, on the
cathode rays, .... . 393
Crookes and the fourth state of matter, .... 394
Objections and alternatives to the charged-particle theory of
cathode rays, ....... 395
Giese's and Schuster's ionic theory of conduction in gases, . . 397
J. J. Thomson measures the velocity of cathode rays, . . 400
Contents. xiii
Page
Discovery of X-rays : hypotheses regarding them, . . 401
Further researches of J. J. Thomson on cathode rays : the ratio m/e, 404
Vitreous and resinous electricity, . . . 406
Determination of the ionic charge by J. J. Thomson, . . 407
Becquerel's radiation : discovery of radio-active substances, . 408
CHAPTER XII.
THE THEORY OF AETHER AND ELECTRONS IN THE CLOSING YEARS
OF THE NINETEENTH CENTURY.
Stokes' theory of aethereal motion near moving bodies, . . 411
Astronomical phenomena in which the velocity of light is involved, 413
Crucial experiments relating to the optics of moving bodies, . 416
Lorentz' theory of electrons, ...... 419
The current of dielectric convection : Rontgen's experiment, . 426
The electronic theory of dispersion, ..... 428
Deduction of Fresnel's formula from the theory of electrons, . 430
Experimental verification of Lorentz' hypothesis, . . . 431
Fitz Gerald's explanation of Michelson's experiment, . . 432
Lorentz' treatise of 1895, . . . . . . . 433
Expression of the potentials in terms of the electronic charges, . 436
Further experiments on the relative motion of earth and aether, . 437
Extension of Lorentz' transformation : Larmor discovers its
connexion with Fitz Gerald's hypothesis of contraction, . 440
Examination of the supposed primacy of the original variables :
fixity relative to the aether : the principle of relativity, . 444
The phenomenon of Zeeman, ..... 449
Connexion of Zeeman's effect with the magnetic rotation of light, . 452
The optical properties of metals, ..... 454
The electronic theory of metals, ..... 456
Thermionics, ........ 464
INDEX, . 470
MEMOKANDUM ON NOTATION.
VECTORS are denoted by letters in clarendon type, as E.
The three components of a vector E are denoted by Ex, Ey, Ez ;
and the magnitude of the vector is denoted by E, so that
The vector product of two vectors E and H, which is denoted
by [E . H], is the vector whose components are (EyHz - E^H^
EZHX - E*HZ, EtHy - EyHx}. Its direction is at right angles to the
direction of E and H, and its magnitude is represented by twice the
area of the triangle formed by them.
The scalar product of E and H is EXHX + EyEy + E^. It is
denoted by (E . H).
OJ^j (1 jjj O Jjj
The quantity — -f — y -I- — is denoted by div E.
The vector whose components are
J — f *t — * ^ . — y _ *\
is denoted by curl E.
If V denote a scalar quantity, the vector whose components are
8F 8F 9F\
- 5T» * ^7' - -5T 1S denoted b7 grad ^
The symbol V is used to denote the vector operator whose
898
components are — , — , — .
dx dy 82
Differentiation with respect to the time is frequently indicated by
a dot placed over the symbol of the variable which is differentiated.
THEORIES OF AETHER AND ELECTRICITY.
CHAPTEK I.
THE THEORY OF THE AETHER IN THE SEVENTEENTH CENTURY.
THE observation of the heavens, which has been pursued con-
tinually from the earliest ages, revealed to the ancients the
regularity of the planetary motions, and gave rise to the
conception of a universal order. Modern research, building on
this foundation, has shown how intimate is the connexion
between the different celestial bodies. They are formed of the
same kind of matter ; they are similar in origin and history ;
and across the vast spaces which divide them they hold
perpetual intercourse.
Until the seventeenth century the only influence which was
known to be capable of passing from star to star was that of
light. Newton added to this the force of gravity ; and it is now
recognized that the power of communicating across vacuous
regions is possessed also by the electric and magnetic attractions.
It is thus erroneous to regard the heavenly bodies as isolated
in vacant space; around and between them is an incessant
conveyance and transformation of energy. To the vehicle of this
activity the name aetlier has been given.
The aether is the solitary tenant of the universe, save for
that infinitesimal fraction of space which is occupied by ordinary
matter. Hence arises a problem which has long engaged
attention, and is not yet completely solved : What relation
subsists between the medium which fills the interstellar void
and the condensations of matter that are scattered throughout
it?
B
$5 l ' r The ^Theory of the • -Aether
The history of this problem may be traced back continuously
to the earlier half of the seventeenth century. It first emerged
clearly in that reconstruction of ideas regarding the physical
universe which was effected by Eene Descartes.
Descartes was born in 1596, the son of Joachim Descartes,
Counsellor to the Parliament of Brittany. As a young man he
followed the profession of arms, and served in the campaigns of
Maurice of Nassau, and the Emperor ; but his twenty-fourth
year brought a profound mental crisis, apparently not unlike
those which have been recorded of many religious leaders ; and
he resolved to devote himself thenceforward to the study of
philosophy.
The age which preceded the birth of Descartes, and that in
which he lived, were marked by events which greatly altered
the prevalent conceptions of the world. The discovery of
America, the circumnavigation of the globe by Drake, the over-
throw of the Ptolemaic system of astronomy, and the invention
of the telescope, all helped to loosen the old foundations and to
make plain the need for a new structure. It was this that
Descartes set himself to erect. His aim was the most ambitious
that can be conceived ; it was nothing less than to create from
the beginning a complete system of human knowledge.
Of such a system the basis must necessarily be metaphysical ;
and this part of Descartes' work is that by which he is most
widely known. But his efforts were also largely devoted to the
mechanical explanation of nature, which indeed he regarded as
one of the chief ends of Philosophy.*
The general character of his writings may be illustrated by
a comparison with those of his most celebrated contemporary, f
Bacon clearly defined the end to be sought for, and laid down
the method by which it was to be attained; then, recognizing
that to discover all the laws of nature is a task beyond the
* Of the works M'hich bear on our present subject, the Dioptrique and the
Me'teores were published at Leyden in 1638, and the Principia Philosophiae at
Amsterdam in 1644, six years before the death of its author.
t The principal philosophical works of Bacon were written about eighteen years
before those of Descartes.
in the SeventeentJi Century. 3
powers of one man or one generation, he left to posterity the
work of filling in the framework which he had designed.
Descartes, on the other hand, desired to leave as little as possible
for his successors to do ; his was a theory of the universe, worked
out as far as possible in every detail. It is, however, impossible
to derive such a theory inductively unless there are at hand
sufficient observational data on which to base the induction ;
and as such data were not available in the age of Descartes,
he was compelled to deduce phenomena from preconceived
principles and causes, after the fashion of the older philosophers.
To the inherent weakness of this method may be traced the
errors that at last brought his scheme to ruin.
The contrast between the systems of Bacon and Descartes is
not unlike that between the Eoman republic and the empire of
Alexander. In the one case we have a career of aggrandizement
pursued with patience for centuries ; in the other a growth of
fungus-like rapidity, a speedy dissolution, and an immense
influence long exerted by the disunited fragments. The
grandeur of Descartes' plan, and the boldness of its execution,
stimulated scientific thought to a degree before unparalleled ;
and it was largely from its ruins that later philosophers
constructed those more valid theories which have endured to
our own time.
Descartes regarded the world as an immense machine,
operating by the motion and pressure of matter. " Give me
matter and motion," he cried, " and I will construct the universe."
A peculiarity which distinguished his system from that which
afterwards sprang from its decay was the rejection of all forms
of action at a distance ; he assumed that force cannot be com-
municated except by actual pressure or impact. By this
assumption he was compelled to provide an explicit mechanism
in order to account for each of the known forces of nature — a
task evidently much more difficult than that which lies before
those who are willing to admit action at a distance as an
ultimate property of matter.
Since the sun interacts with the planets, in sending them
B 2
4 The Theory of the Aether
light and heat and influencing their motions, it followed from
Descartes' principle that interplanetary space must be a plenum,,
occupied by matter imperceptible to the touch but capable of
serving as the vehicle of force and light. This conclusion in
turn determined the view which he adopted on the all- important
question of the nature of matter.
Matter, in the Cartesian philosophy, is characterized not by
impenetrability, or by any quality recognizable by the senses,,
but simply by extension ; extension constitutes matter, and
matter constitutes space. The basis of all things is a primitive,,
elementary, unique type of matter, boundless in extent and
infinitely divisible. In the process of evolution of the universe
three distinct forms of this matter have originated, correspond-
ing respectively to the luminous matter of the sun, the
transparent matter of interplanetary space, and the dense,
opaque matter of the earth. " The first is constituted by what
has been scraped off the other particles of matter when they
were rounded ; it moves with so much velocity that when it
meets other bodies the force of its agitation causes it to be
broken and divided by them into a heap of small particles that
are of such a figure as to fill exactly all the holes and small
interstices which they find around these bodies. The next type
includes most of the rest of matter ; its particles are spherical,
and are very small compared with the bodies we see on the
earth ; but nevertheless they have a finite magnitude, so that
they can be divided into others yet smaller. There exists in
addition a third type exemplified by some kinds of matter —
namely, those which, on account of their size and figure, cannot be
so easily moved as the preceding. I will endeavour to show that
all the bodies of the visible world are composed of these three
forms of matter, as of three distinct elements ; in fact, that the sun
and the fixed stars are formed of the first of these elements, the
interplanetary spaces of the second, and the earth, with the
planets and comets, of the third. For, seeing that the sun and
the fixed stars emit light, the heavens transmit it, and the earth,
the planets, and the comets reflect it, it appears to me that there
in the Seventeenth Century. 5
is ground for using these three qualities of luminosity, trans-
parence, and opacity, in order to distinguish the three elements
of the visible world.*
According to Descartes' theory, the sun is the centre of an
immense vortex formed of the first or subtlest kind of inatter.f
The vehicle of light in interplanetary space is matter of the
second kind or element, composed of a closely packed assemblage
of globules whose size is intermediate between that of the
vortex-matter and that of ponderable matter. The globules of
the second element, and all the matter of the first element, are
constantly straining away from the centres around which they
turn, owing to the centrifugal force of the vortices ;J so that the
globules are pressed in contact with each other, and tend to
move outwards, although they do not actually so move.§ It is
the transmission of this pressure which constitutes light ; the
action of light therefore extends on all sides round the sun and
fixed stars, and travels instantaneously to any distance. |j In
the Dwptrique$ vision is compared to the perception of the
presence of objects which a blind man obtains by the use of his
stick ; the transmission of pressure along the stick from the
object to the hand being analogous to the transmission of
pressure from a luminous object to the eye by the second kind
of matter.
Descartes supposed the " diversities of colour and light " to
he due to the different ways in which the matter moves.** In
the Meteores,^ the various colours are connected with different
rotatory velocities of the globules, the particles winch rotate most
rapidly giving the sensation of red, the slower ones of yellow, and
the slowest of green and blue — the order of colours being taken
from the rainbow. The assertion of the dependence of colour
* Principia, Part iii, § 52.
t It is curious to speculate on the impression which would have been produced
had the spirality of nehulse heen discovered hefore the overthrow of the Cartesian
theory of vortices.
J Ibid., §§ 55-59. § Ibid., § 63. || Ibid., § 64. IT Discours premier.
** Principia, Part iv, § 195. ft Discours Huitieme.
6 The Theory of the Aether
on periodic time is a curious foreshadowing of one of the
great discoveries of Newton.
The general explanation of light on these principles was
amplified by a more particular discussion of reflexion and
refraction. The law of reflexion— that the angles of incidence
and refraction are equal — had been known to the Greeks ; but
the law of refraction — that the sines of the angles of incidence
and refraction are to each other in a ratio depending on the
media — was now published for the first time.* Descartes gave
it as his own ; but he seems to have been under considerable
obligations to Willebrord Snell (b. 1591, d. 1626), Professor of
Mathematics at Leyden, who had discovered it experimentally
(though not in the form in which Descartes gave it) about
1621. Snell did not publish his result, but communicated it in
manuscript to several persons, and Huygens affirms that this
manuscript had been seen by Descartes.
Descartes presents the law as a deduction from theory.
This, however, he is able to do only by the aid of analogy ;.
when rays meet ponderable bodies, " they are liable to be
deflected or stopped in the same way as the motion of a ball or
a stone impinging 011 a body " ; for " it is easy to believe that
the action or inclination to move, which I have said must be
taken for light, ought to follow in this the same laws as
motion."f Thus he replaces light, whose velocity of propagation
he believes to be always infinite, by a projectile whose velocity
varies from one medium to another. The law of refraction is
then proved as follows J : —
Let a ball thrown from A meet at B a cloth CBE, so weak
that the ball is able to break through it and pass beyond, but
with its resultant velocity reduced in some definite proportion,,
say 1 : k.
Then if BI be a length measured on the refracted ray
equal to AB, the projectile will take k times as long to
describe BI as it took to describe AB. But the component
* Dioptrique, Discount second. t Jbid., Discows premier.
% Ibid., Discotirs second.
in the Seventeenth Century. 7
of velocity parallel to the cloth must be unaffected by the
impact; and therefore the projection BE of the refracted ray
must be k times as long as the projection BC of the incident
I
ray. So if i and r denote the angles of incidence and refraction,
we have
• BE BC
or the sines of the angles of incidence and refraction are in a
constant ratio ; this is the law of refraction.
Desiring to include all known phenomena in .his system,
Descartes devoted some attention to a class of effects which
were at that time little thought of, but which were destined to
play a great part in the subsequent development of Physics.
The ancients were acquainted with the curious properties
possessed by two minerals, amber (riXtKrpov) and magnetic
iron ore (77 \iOos Mayv?}r/e). The former, when rubbed,
attracts light bodies : the latter has the power of attracting
iron.
The use of the magnet for the purpose of indicating direc-
tion at sea does not seem to have been derived from classical
antiquity ; but it was certainly known in the time of the
Crusades. Indeed, magnetism was one of the few sciences
which progressed during the Middle Ages ; for in the thirteenth
century Petrus Peregrinus,* a native of Maricourt in Picardy,
made a discovery of fundamental importance.
Taking a natural magnet or lodestone, which had been
rounded into a globular form, he laid it on a needle, and marked
* His Epistola was written in 1269.
8 The Theory of the Aether
the line along which the needle set itself. Then laying the
needle on other parts of the stone, he obtained more lines in
the same way. When the entire surface of the stone had been
covered with such lines, their general disposition became evident;
they formed circles, which girdled the stone in exactly the same
way as meridians of longitude girdle the earth ; and there were
two points at opposite ends of the stone through which all the
circles passed, just as all the meridians pass through the Arctic
and Antarctic poles of the earth.* Struck by the analogy,
Peregrinus proposed to call these two points the poles of the
magnet : and he observed that the way in which magnets set
themselves and attract each other depends solely on the position
of their poles, as if these were the seat of the magnetic power.
Such was the origin of those theories of poles and polarization
which in later ages have played so great a part in Natural
Philosophy.
The observations of Peregrinus were greatly extended not
long before the tune of Descartes by William Gilberd or Gilbertf
(6. 1540, d. 1603). Gilbert was born at Colchester: after
studying at Cambridge, he took up medical practice in London,
and had the honour of being appointed physician to Queen
Elizabeth. In 1600 he published a work* on Magnetism and
Electricity, with which the modern history of both subjects
begins.
Of Gilbert's electrical researches we shall speak later : in
magnetism he made the capital discovery of the reason why
magnets set in definite orientations with respect to the earth ;
which is, that the earth is itself a great magnet, having one of
its poles in high northern and the other in high southern
latitudes. Thus the property of the compass was seen to be
included in the general principle, that the north-seeking pole of
* " Procul dubio oranes lineae hujusmodi in duo puncta concurrent sicut omnes
orbes meridian! in duo concurrunt polos mundi oppositos."
t The form in the Colchester records is Gilberd.
J Gulielmi Gilberti de Magnete, Magneticisque corporibus, et de magno magnete
tellure : London, 1600. An English translation by P. F. Mottelay was published
in 1893.
in the Seventeenth Century. 9
every magnet attracts the south-seeking pole of every other
magnet, and repels its north-seeking pole.
Descartes attempted* to account for magnetic phenomena
by his theory of vortices. A vortex of fluid matter was
postulated round each magnet, the matter of the vortex entering
by one pole and leaving by the other : this matter was supposed
to act on iron and steel by virtue of a special resistance to its
motion afforded by the molecules of those substances.
Crude though the Cartesian system was in this and many
other features, there is no doubt that by presenting definite
conceptions of molecular activity, and applying them to so wide
a range of phenomena, it stimulated the spirit of inquiry, and
prepared the way for the more accurate theories that came after.
In its own day it met with great acceptance: the confusion which
had resulted from the destruction of the old order was now, as
it seemed, ended by a reconstruction of knowledge in a system
at once credible and complete. Nor did its influence quickly
wane ; for even at Cambridge it was studied long after Newton
had published his theory of gravitation ;f and in the middle of
the eighteenth century Euler and two of the Bernoullis based
the explanation of magnetism on the hypothesis of vertices.*
Descartes' theory of light rapidly displaced the conceptions
which had held sway in the Middle Ages. The validity
of his explanation of refraction was, however, called in
question by his fellow-countryman Pierre de Ferinat (b. 1601,
d. 1665),§ and a controversy ensued, which was kept up
by the Cartesians long after the death of their master. Fermat
* Principia, Part iv, § 133 sqq.
•f Winston has recorded that, having returned to Cambridge after his
ordination in 1693, he resumed his studies there, " particularly the Mathematicks,
and the Cartesian Philosophy : which was alone in Vogue with us at that Time.
But it was not long before I, with immense Pains, but no Assistance, set myself
with the utmost Zeal to the study of Sir Isaac Newton's M-onderful Discoveries."
— \Vhiston's Memoirs (1749), i, p. 36.
J Their memoirs shared a prize of the French Academy in 1743, and were
printed in 1752 in the Heciieil des pieces qui ontremporte les prix de VAcad., tome v.
§ Renati Descartes Epistolae, Pars tertia ; Amstelodami, 1683. The Fennat
correspondence is comprised in letters xxix to XLVI.
10 The Theory of the Aether
eventually introduced a new fundamental law, from which he
proposed to deduce the paths of rays of light. This was the
celebrated Principle of Least Time, enunciated* in the form,
" Nature always acts by the shortest course." From it the law
of reflexion can readily be derived, since the path described by
light between a point 011 the incident ray and a point on the
reflected ray is the shortest possible consistent with the con-
dition of meeting the reflecting surfaces. t In order to obtain the
law of refraction, Fermat assumed that " the resistance of the
media is different," and applied his "method of maxima and
minima " to find the path which would be described in the least
time from a point of one medium to a point of the other. In
1661 he arrived at the solution.* "The result of my work," he
writes, " has been the most extraordinary, the most unforeseen,
and the happiest, that ever was ; for, after having performed all
the equations, multiplications, antitheses, and other operations
of my method, and having finally finished the problem, I have
found that my principle gives exactly and precisely the same
proportion for the refractions which Monsieur Descartes has
established." His surprise was all the greater, as he had
supposed light to move more slowly in dense than in rare media,
whereas Descartes had (as will be evident from the demonstration
given above) been obliged to make the contrary supposition.
Although Fermat's result was correct, and, indeed, of high
permanent interest, the principles from which it was derived
were metaphysical rather than physical in character, and con-
sequently were of little use for the purpose of framing a
mechanical explanation of light. Descartes' theory therefore
held the field until the publication in 1667§ of the Micrographics
* Epist. XLII, written at Toulouse in August, 1657, to Monsieur de la
Chambre ; reprinted in (Euvres de Fermat (ed. 1891), ii, p. 354.
t That reflected light follows the shortest path was no new result, for it had
been affirmed (and attributed to Hero of Alexandria) in the Ke<t>aA.cua rwv OTTTIKUHT
of Heliodorns of Larissa, a work of which several editions were published in the
seventeenth, century.
J Epist. XLIII, written at Toulouse on Jan. 1, 1662 ; reprinted in (Euvres de
Fermat, ii, p. 457 ; i, pp. 170, 173.
§ The imprimatur of Viscount Brouncker, P.R.S., is dated Nov. 23, 1664.
in the Seventeenth Centnry. 11
of Eobert Hooke (b. 1635, d. 1703), one of the founders of the
Eoyal Society, and at one time its Secretary.
Hooke, who was both an observer and a theorist, made two
experimental discoveries which concern our present subject ; but
in both of these, as it appeared, he had been anticipated. The
first* was the observation of the iridescent colours which are
seen when light falls on a thin layer of air between two glass
plates or lenses, or on a thin film of any transparent substance.
These are generally known as the " colours of thin plates," or
" Newton's rings " ; they had been previously observed by Boyle.f
Hooke's second experimental discovery,^ made after the date of
the Micrographia, was that light in air is not propagated exactly
in straight lines, but that there is some illumination within the
geometrical shadow of an opaque body. This observation had
been published in 1665 in. a posthumous work§ of Francesco
Maria Grimaldi (b. 1618, d. 1663), who had given to the phe-
nomenon the name diffraction.
Hooke's theoretical investigations on light were of great
importance, representing as they do the transition from the
Cartesian system to the fully developed theory of undulations.
He begins by attacking Descartes' proposition, that light is a
tendency to motion rather than an actual motion. " There is,"
he observes, 1 1 " no luminous Body but has the parts of it in
motion more or less " ; and this motion is " exceeding quick."
Moreover, since some bodies (e.g. the diamond when rubbed or
heated in the dark) shine for a considerable time without being
wasted away, it follows that whatever is in motion is not per-
manently lost to the body, and therefore that the motion must
be of a to-and-fro or vibratory character. The amplitude of the
vibrations must be exceedingly small, since some luminous bodies
(e.g. the diamond again) are very hard, and so cannot yield or
bend to any sensible extent.
* Micrographia, p. 47. t Boyle's Works (ed. 1772), i, p. 742.
% Hooke's Posthumous Works, p. 186.
§ Pkysico- Mathesis de lumine, coloribits, et iride. Bologna, 1665 ; book i, prop. i.
|| Micrographia, p. 55.
12 The Theory of the Aether
Concluding, then, that the condition associated with the
emission of light by a luminous body is a rapid vibratory motion
of very small amplitude, Hooke next inquires how light travels
through space. " The next thing we are to consider," he says,
" is the way or manner of the trajection of this motion through
the interpos'd pellucid body to the eye : And here it will be
easily granted —
" First, that it must be a body susceptible and impartible of
this motion that will deserve the name of a Transparent ; and
next, that the parts of such a body must be homogeneous, or of
the same kind.
" Thirdly, that the constitution and motion of the parts must
be such that the appulse of the luminous body may be commu-
nicated or propagated through it to the greatest imaginable
distance in the least imaginable time, though I see no reason to
affirm that it must be in an instant.
" Fourthly, that the motion is propagated every way through
an Homogeneous medium by direct or straight lines extended every
way like Eays from the centre of a Sphere.
" Fifthly, in an Homogeneous medium this motion is propa-
gated every way with equal velocity, whence necessarily every
pulse or vibration of the luminous body will generate a Sphere,
which will continually increase, and grow bigger, just after the
same manner (though indefinitely swifter) as the waves or rings
on the surface of the water do swell into bigger and bigger
circles about a point of it, where by the sinking of a Stone the
motion was begun, whence it necessarily follows, that all the
parts of these Spheres undulated through an Homogeneous medium
cut the Kays at right angles."
Here we have a fairly definite mechanical conception. It
resembles that of Descartes in postulating a medium as the
vehicle of light ; but according to the Cartesian hypothesis the
disturbance is a statical pressure in this medium, while in
Hooke's theory it is a rapid vibratory motion of small amplitude.
In the above extract Hooke introduces, moreover, the idea of
the wave-swrface, or locus at any instant of a disturbance gene-
in the Seventeenth Century. 13
rated originally at a point, and affirms that it is a sphere,
whose, centre is the point in question, and whose radii are
the rays of light issuing from the point.
Hooke's next effort was to produce a mechanical theory of
refraction, to replace that given by Descartes. " Because," he
says, "all transparent mediums are not Homogeneous to one
another, therefore we will next examine how this pulse or motion
will be propagated through differingly transparent mediums.
And here, according to the most acute and excellent Philosopher
Des Cartes, I suppose the sine of the angle of inclination in the
first medium to be to the sine of refraction in the second, as the
density of the first to the density of the second. By density, I
mean not the density in respect of gravity (with which the
refractions or transparency of mediums hold no proportion), but
in respect only to the trajeetion of the Kays of light, in which
respect they only differ in this, that the one propagates the
pulse more easily and weakly, the other more slowly, but
more strongly. But as for the pulses themselves, they will
by the refraction acquire another property, which we shall now
endeavour to explicate.
"We will suppose, therefore, in the first Figure, ACFD to be
a physical Kay, or ABC and DEFto be two mathematical Kaysr
trajected from a very remote point of a luminous body through
14 The Theory of the Aether
an Homogeneous transparent medium LL, and DA, EB, FC, to be
small portions of the orbicular impulses which must therefore
cut the Rays at right angles : these Rays meeting with the plain
surface NO of a medium that yields an easier transitus to the
propagation of light, and falling obliquely on it, they will in the
medium MM be refracted towards the perpendicular of the
surface. And because this medium is more easily trajected than
the former by a third, therefore the point 0 of the orbicular
pulse FG will be moved to If four spaces in the same time that
F, the other end of it, is moved to three spaces, therefore the
whole refracted pulse to H shall be oblique to the refracted Rays
GHK and £/."
Although this is not in all respects successful, it represents
a decided advance on the treatment of the same problem by
Descartes, which rested on a mere analogy. Hooke tries to
determine what happens to the wave-front when it meets
the interface between two media ; and for this end he intro-
duces the correct principle that the side of the wave-front
which first meets the interface will go forward in the second
medium with the velocity proper to that medium, while the
other side of the wave-front which is still in the first medium
is still moving with the old velocity : so that the wave-front
will be deflected in the transition from one medium to the
other.
This deflection of the wave-front was supposed by Hooke to
be the origin of the prismatic colours. He regarded natural or
white light as the simplest type of disturbance, being consti-
tuted by a simple and uniform pulse at right angles to the
direction of propagation, and inferred that colour is generated
by the distortion to which this disturbance is subjected in the
process of refraction. "The Ray,"* he says, " is dispersed, split, and
opened by its Refraction at the Superficies of a second medium,
and from a line is opened into a diverging Superficies, and
so obliquated, whereby the appearances of Colours are produced."
* Hooke, Posthnmo/is Works, p. 82.
in the Seventeenth Century. \ 5
" Colour/' he says in another place,* " is nothing but the
disturbance of light by the communication of the pulse to other
transparent mediums, that is by the refraction thereof." His
precise hypothesis regarding the different colours wasf "that
Blue is an impression on the Retina of an oblique and confus'd
pulse of light, whose weakest part precedes, and whose
strongest follows. And, that red is an impression on the Retina
of an oblique and confus'd pulse of light, whose strongest part
precedes, and whose weakest follows."
Hooke's theory of colour was completely overthrown, within
a few years of its publication, by one of the earliest discoveries
of Isaac Xewton (b. 1642, d. 1727). Newton, who was elected
a Fellow of Trinity College, Cambridge, in 1667, had in the
beginning of 1666 obtained a triangular prism, " to try-
therewith the celebrated Phaenomena of Colours." For this
purpose, " having darkened my chamber, and made a small hole
in my window-shuts, to let in a convenient quantity of the
Sun's light, I placed my Prisme at his entrance, that it might
be thereby refracted to the opposite wall. It was at first a
very pleasing divertisement, to view the vivid and intense
colours produced thereby ; but after a while applying myself to
consider them more circumspectly, I became surprised to see
them in an oblong form, which, according to the received laws
of Refraction, I expected should have been circular" The
length of the coloured spectrum was in fact about five times as
great as its breadth.
This puzzling fact he set himself to study ; and after more
experiments the true explanation was discovered — namely,
that ordinary white light is really a mixture of rays of every
variety of colour, and that the elongation of the spectrum is
due to the differences in the refractive power of the glass for
these different rays.
" Amidst these thoughts," he tells us,+ " I was forced from
*To the Royal Society, February 15, 1671-2.
t Micrographia, p. 64.
J Phil. Trans., Xo. 80, February 19, 1671-2.
16 The Theory of the Aether
Cambridge by the intervening Plague " ; this was in 1666, and
his memoir on the subject was not presented to the Koyal
Society until five years later. In it he propounds a theory of
colour directly opposed to that of Hooke. " Colours," he says,
"are not Qualifications of light derived from Refractions, or
Reflections of natural Bodies (as 'tis generally believed), but
Original and connate properties, which in divers Rays are divers.
Some Rays are disposed to exhibit a red colour and no other :
some a yellow and no other, some a green and no other, and so
of the rest. Nor are there only Rays proper and particular to
the more eminent colours, but even to all their intermediate
gradations.
" To the same degree of Refrangibility ever belongs the
same colour, and to the same colour ever belongs the same
degree of Refrangibility."
" The species of colour, and degree of Refrangibility proper
to any particular sort of Rays, is not mutable by Refraction, nor
by Reflection from natural bodies, nor by, any other cause, that
I could yet observe. When any one sort of Rays hath been
well parted from those of other kinds, it hath afterwards
obstinately retained its colour, notwithstanding my utmost
endeavours to change it."
The publication of the new theory gave rise to an acute
controversy. As might have been expected, Hooke was foremost
among the opponents, and led the attack with some degree of
asperity. When it is remembered that at this time Newton
was at the outset of his career, while Hooke was an older man,
with an established reputation, such harshness appears par-
ticularly ungenerous; and it is likely that the unpleasant
consequences which followed the announcement of his first
great discovery had much to do with the reluctance which
Newton ever afterwards showed to publish his results to the
world.
In the course of the discussion Newton found occasion to
explain more fully the views which he entertained regarding
the nature of light. Hooke charged him with holding the
in the Seventeenth Century. 17
doctrine that light is a material substance. Now Newton had, as
a matter of fact, a great dislike of the more imaginative kind of
hypotheses ; he altogether renounced the attempt to construct
the universe from its foundations after the fashion of Descartes,
and aspired to nothing more than a formulation of the laws
which directly govern the actual phenomena. His theory of
gravitation, for example, is strictly an expression of the results
of observation, and involves no hypothesis as to the cause of the
attraction which subsists between ponderable bodies ; and his
own desire in regard to optics was to present a theory free from
speculation as to the hidden mechanism of light. Accordingly,
in reply to Hooke's criticism, he protested* that his views on
colour were in no way bound up with any particular conception
of the ultimate nature of optical processes.
Xewton was, however, unable to carry out his plan of
connecting together the phenomena of light into a coherent
and reasoned whole without having recourse to hypotheses. The
hypothesis of Hooke, that light consists in vibrations of an
aether, he rejected for reasons which at that time were perfectly
cogent, and which indeed were not successfully refuted for over
a century. One of these was the incompetence of the wave-
theory to account for the rectilinear propagation of light, and
another was its inability to embrace the facts — discovered, as
we shall presently see, by Huygens, and first interpreted
correctly by Newton himself — of polarization. On the whole,
he seems to have favoured a scheme of which the following may
be taken as a summaryf : —
All space is permeated by an elastic medium or aether, which
is capable of propagating vibrations in the same way as the
*Phil. Trans, vii, 1672, p. 5086.
t Cf. Newton's memoir in Phil. Trans, vii, 1672 ; his memoir presented to the
Royal Society in December, 1675, which is printed in Birch, iii, p. 247; his
Opticks, especially Queries 18, 19, 20, 21, 23, 29; the Scholium at the end of
the Principia ; and a letter to Boyle, written in February, 1678-9, which is printed
in Horsley's Newtoni Opera, p. 385.
In the Principia, Book I., section xiv, the analogy between rays of light and
streams of corpuscles is indicated ; but Newton does not commit himself to any
theory of light based on this.
C
18 The Theory of the Aether
air propagates the vibrations of sound, but with far greater
velocity.
This aether pervades the pores of all material bodies, and
is the cause of their cohesion ; its density varies from one body
to another, being greatest in the free interplanetary spaces. It
is not necessarily a single uniform substance : but just as air
contains aqueous vapour, so the aether may contain various
" aethereal spirits," adapted to produce the phenomena of
electricity, magnetism, and gravitation.
The vibrations of the aether cannot, for the reasons already
mentioned, be supposed in themselves to constitute light.
Light is therefore taken to be " something of a different kind,
propagated from lucid bodies. They, that will, may suppose
it an aggregate of various peripatetic qualities. Others may
suppose it multitudes of unimaginable small and swift
corpuscles of various sizes, springing from shining bodies
at great distances one after another; but yet without any
sensible interval of time, and continually urged forward by a
principle of motion, which in the beginning accelerates them,
till the resistance of the aethereal medium equals the force of
that principle, much after the manner that bodies let fall in
water are accelerated till the resistance of the water equals the
force of gravity. But they, that like not this, may suppose
light any other corporeal emanation, or any impulse or motion
of any other medium or aethereal spirit diffused through the
main body of aether, or what else they can imagine proper for
this purpose. To avoid dispute, and make this hypothesis
general, let every man here take his fancy ; only whatever
light be, I suppose it consists of rays differing from one another
in contingent circumstances, as bigness, form, or vigour."*
In any case, light and aether are capable of mutual inter-
action; aether is in fact the intermediary between light and
ponderable matter. When a ray of light meets a stratum of
aether denser or rarer than that through which it has lately
been passing, it is, in general, deflected from its rectilinear
* Royal Society, Dec. 9, 1675.
in the Seventeenth Century. 19
course ; and differences of density of the aether between one
material medium and another account on these principles for
the reflexion and refraction of light. The condensation or
rarefaction of the aether due to a material body extends to
some little distance from the surface of the body, so that the
inflexion due to it is really continuous, and not abrupt; and
this further explains diffraction, which Newton took to be
" only a new kind of refraction, caused, perhaps, by the
external aethers beginning to grow rarer a little before it
came at the opake body, than it was in free spaces."
Although the regular vibrations of Newton's aether were not
supposed to constitute light, its irregular turbulence seems to
have represented fairly closely his conception of heat. He
supposed that when light is absorbed by a material body,
vibrations are set up in the aether, and are recognizable as
the heat which is always generated in such cases. The
conduction of heat from hot bodies to contiguous cold ones he
conceived to be effected by vibrations of the aether propagated
between them ; and he supposed that it is the violent agitation
of aethereal motions which excites incandescent substances to
emit light.
Assuming with Newton that light is not actually con-
stituted by the vibrations of an aether, even though such
vibrations may exist in close connexion with it, the most
definite and easily conceived supposition is that rays of light
are streams of corpuscles emitted by luminous bodies. Although
this was not the hypothesis of Descartes himself, it was so
thoroughly akin to his general scheme that the scientific men
of Newton's generation, who were for the most part deeply
imbued with the Cartesian philosophy, instinctively selected
it from the wide choice of hypotheses which Newton had offered
them ; and by later writers it was generally associated with
Newton's name. A curious argument in its favour was drawn
from a phenomenon which had then been known for nearly half
a century : Vincenzo Cascariolo, a shoemaker of Bologna, had
discovered, about 1630, that a substance, which afterwards
C 2
20 The Theory of the Aether
received the name of Bologna stone or Bologna phosphorus, has-
the property of shining in the dark after it has been exposed
for some time to sunlight ; and the storage of light which
seemed to be here involved was more easily explicable on the
corpuscular theory than on any other. The evidence in
this quarter, however, pointed the other way when it was
found that phosphorescent substances do not necessarily emit
the same kind of light as that which was used to stimulate
them.
In accordance with his earliest discovery, Newton considered
colour to be an inherent characteristic of light, and inferred
that it must be associated with some definite quality of the
corpuscles or aether-vibrations. The corpuscles corresponding
to different colours would, he remarked, like sonorous bodies of
different pitch, excite vibrations of different types in the
aether ; and " if by any means those [aether- vibrations] of
unequal bignesses be separated from one another, the largest
beget a Sensation of a Red colour, the least or shortest of a
deep Violet, and the intermediate ones, of intermediate colours ;
much after the manner that bodies, according to their several
sizes, shapes, and motions, excite vibrations in the Air of various
bignesses, which, according to those bignesses, make several
Tones in Sound."*
This sentence is the first enunciation of the great principle
that homogeneous light is essentially periodic in its nature, and
that differences of period correspond to differences of colour.
The analogy with Sound is obvious ; and it may be remarked
in passing that Newton's theory of periodic vibrations in an
elastic medium, which he developed! in connexion with the
explanation of Sound, would alone entitle him to a place among
those who have exercised the greatest influence on the theory
of light, even if he had made no direct contribution to the
latter subject.
* Phil. Trans, vii (1672), p. 5088.
t Newton's Prmcipia, Book ii., Props, xliii.-l.
in the Seventeenth Century. 21
Newton devoted considerable attention to the colours of
thin, plates, and determined the empirical laws of the
phenomena with great accuracy. In order to explain them, he
supposed that " every ray of light, in its passage through any
refracting surface, is put into a certain transient constitution or
state, which, in the progress of the ray, returns at equal
intervals, and disposes the ray, at every return, to be easily
transmitted through the next refracting surface, and, between
the returns, to be easily reflected by it."* The interval
between two consecutive dispositions to easy transmission, or
•" length of fit," he supposed to depend on the colour, being
greatest for red light and least for violet. If then a ray of
homogeneous light falls on a thin plate, its fortunes as regards
transmission and reflexion at the two surfaces will depend on
the relation which the length of fit bears to the thickness of
the plate ; and on this basis he built up a theory of the colours
of thin plates. It is evident that Newton's "length of fit"
corresponds in some measure to the quantity which in the
undulatory theory is called the wave-length of the light ; but
the suppositions of easy transmission and reflexion were soon
found inadequate to explain all Newton's experimental results —
.at least without making other and more complicated additional
assumptions.
At the time of the publication of Hooke's Micrographia, and
Newton's theory of colours, it was not known whether light
is propagated instantaneously or not. An attempt to settle
the question experimentally had been made many years
previously by Galileo,f who had stationed two men with
lanterns at a considerable distance from each other ; one of
them was directed to observe when the other uncovered his
light, and exhibit his own the moment he perceived it. But
the interval of time required by the light for its journey was
too small to be perceived in this way ; and the discovery was
* Optic ks, Book ii., Prop. 12.
t Discorri e dimostrazioiti matemaliche, p. 43 of the Elzevir edition of 1638.
22 The Theory of the Aether
ultimately made by an astronomer. It was observed in 1675
by Olof Roemer* (b. 1644, d. 1710) that the eclipses of the first
satellites of Jupiter were apparently affected by an unknown
disturbing cause ; the time of the occurrence of the phenomenon
was retarded when the earth and Jupiter, in the course of their
orbital motions, happened to be most remote from each other,
and accelerated in the contrary case. Eoemer explained this
by supposing that light requires a finite time for its pro-
pagation from the satellite to the earth ; and by observations of
eclipses, he calculated the interval required for its passage from
the sun to the earth (the light-equation, as it is called) to be
11 minutes, f
Shortly after Roemer's discovery, the wave-theory of light
was greatly improved and extended by Christiaan Huygens
(b. 1629, d. 1695). Huygens, who at the time was living in
Paris, communicated his results in 1678 to Cassini, Eoemer,
De la Hire, and the other physicists of the French Academy,
and prepared a manuscript of considerable length on the subject.
This he proposed to translate into Latin, and to publish in that
language together with a treatise on the Optics of Telescopes ;
but the work of translation making little progress, after a delay
of twelve years, he decided to print the work on wave-theory
in its original form. In 1690 it appeared at Ley den, J under
the title Traite de la lumiere ou sont expliquees les causes de ce
qui luy arrive dans la reflexion et dans la refraction. Et parti-
*Mem. de 1'Acad. x. (1666-1699), p. 575.
t It was soon recognized that Roemer's value was too large ; and the
astronomers of the succeeding half-century reduced it to 7 minutes. Delambre,
by an investigation whose details appear to have been completely destroyed,
published in 1817 the value 493 -2s, from a discussion of eclipses of Jupiter's
satellites during the previous 150 years. Glasenapp, in an inaugural dissertation
published in 1875, discussed the eclipses of the first satellite between 1848 and
1870, and derived, by different assumptions, values between 496s and 501s, the
most probable value being 500-88. Sampson, in 1909, derived 498'64S from his
own readings of the Harvard Observations, and 498'79S from the Harvard readings,
with probable errors of about + 0'02". The inequalities of Jupiter's surface give
rise to some difficulty in exact determinations.
% Huygens had by this time returned to Holland.
in the Seventeenth Century. 23
culierement dans Vetrange refraction du cristal d'Islande. Par
C.ff.D.Z*
The truth of Hooke's hypothesis, that light is essentially a
form of motion, seemed to Huygens to be proved ]}y the effects
observed with burning-glasses ; for in the combustion induced at
the focus of the glass, the molecules of bodies are dissociated ;
which, as he remarked, must be taken as a certain sign of motion,
if, in conformity to the Cartesian philosophy, we seek the cause
of all natural phenomena in purely mechanical actions.
The question then arises as to whether the motion is that
of a medium, as is supposed in Hooke's theory, or whether it
may be compared rather to that of a flight of arrows, as in the
corpuscular theory. Huygens decided that the former alter-
native is the only tenable one, since beams of light proceeding
in directions inclined to each other do not interfere with each
other in any way.
Moreover, it had previously been shown by Torricelli that
light is transmitted as readily through a vacuum as through
air ; and from this Huygens inferred that the medium or aether
in which the propagation takes place must penetrate all matter,
and be present even in all so-called vacua.
The process of wave-propagation he discussed by aid of a
principle which was nowf introduced for the first time, and has
since been generally known by his name. It may be stated
thus : Consider a wave-front,* or locus of disturbance, as it
exists at a definite instant t0 : then each surface-element of the
wave-front may be regarded as the source of a secondary wave,
which in a homogeneous isotropic medium will be propagated
outwards from the surface-element in the form of a sphere
whose radius at any subsequent instant t is proportional to
(t-t0) ; and the wave-front which represents the whole distur-
* i.e. Cbristiaan Huygens de Zuylichem. The custom of indicating names by
initials was not unusual in that age.
t Traite de la lum., p. 17.
I It maybe remarked that Huygens' " waves " are really what modern writers,
following Hooke, call " pulses "; Huygens never considered true wave-trains
having the property of periodicity.
24 The Theory of the Aether
bance at the instant t is simply the envelope of the secondary
waves which arise from the various surface elements of the
original wave-front.* The introduction of this principle enabled
Huygens to succeed where Hooke and other contemporary
wave-theoristsf had failed, in achieving the explanation of
refraction and reflexion. His method was to combine his own
principle with Hooke's device of following separately the fortunes
of the right-hand and left-hand sides of a wave-front when it
reaches the interface between two media. The actual explana-
tion for the case of reflexion is as follows : —
Let AB represent the interface at which reflexion takes
place, AHC the incident wave-front at an instant £0, GMB the
position which the wave-front would occupy at a later instant t
if the propagation were not interrupted by reflexion. Then by
"G
Huygens' principle the secondary wave from A is at the instant
t a sphere ENS of radius equal to AG : the disturbance from Ht
after meeting the interface at K, will generate a secondary
wave TV oi radius equal to KM, and similarly the secondary
wave corresponding to any other element of the original wave-
* The justification for this was given long afterwards by Fresnel, Annales de
chimie, xxi.
t e.g. Ignace Gaston Pardies and Pierre Ango, the latter of whom published
a work on Optics at Paris'in 1682.
in the Seventeenth Century. 25
front can be found. It is obvious that the envelope of these
secondary waves, which constitutes the final wave-front, will be
a plane BN, which will be inclined to AB at the same angle as
AC. This gives the law of reflexion.
The law of refraction is established by similar reasoning,
on the supposition that the velocity of light depends on the
medium in which it is propagated. Since a ray which passes
from air to glass is bent inwards towards the normal, it may be
inferred that light travels more slowly in glass than in air.
Huygens offered a physical explanation of the variation in
velocity of light from one medium to another, by supposing
that transparent bodies consist of hard particles which interact
with the aethereal matter, modifying its elasticity. The
opacity of metals he explained by an extension of the same
idea, supposing that some of the particles of metals are hard
(these account for reflexion) and the rest soft : the latter destroy
the luminous motion by damping it.
The second half of the Theorie de la lumiere is concerned with
a phenomenon which had been discovered a few years pre-
viously by a Danish philosopher, Erasmus Bartholin (b. 1625,
d. 1698). A sailor had brought from Iceland to Copenhagen a
number of beautiful crystals which he had collected in the Bay
of Eoerford. Bartholin, into whose hands they passed, noticed*
that any small object viewed through one of these crystals
appeared double, and found the immediate cause of this in the
fact that a ray of light entering the crystal gave rise in general
to two refracted rays. One of these rays was subject to the
ordinary law of refraction, while the other, which was called
the extraordinary ray, obeyed a different law, which Bartholin
did not succeed in determining.
The matter had arrived at this stage when it was taken up
by Huygens. Since in his conception each ray of light corresponds
to the propagation of a wave-front, the two rays in Iceland
spar must correspond to two different wave-fronts propagated
* Ejcperimenta cristatti Islandici disdiaclastici : 1669.
26 The Theory of the Aether
simultaneously. In this idea he found no difficulty ; as he says :
" It is certain that a space occupied by more than one kind of
matter may permit the propagation of several kinds of waves,
different in velocity; for this actually happens in air mixed
with aethereal matter, where sound-waves and light- waves are
propagated together."
Accordingly he supposed that a light-disturbance generated
at any spot within a crystal of Iceland spar spreads out in the
form of a wave-surface, composed of a sphere and a spheroid
having the origin of disturbance as centre. The spherical wave-
front corresponds to the ordinary ray, and the spheroid to the
extraordinary ray ; and the direction in which the extraordinary
ray is refracted may be determined by a geometrical construc-
tion, in which the spheroid takes the place which in the
ordinary construction is taken by the sphere.
Thus, let the plane of the figure be at right angles to the
intersection of the wave-front with the surface of the crystal ;
let AB represent the trace of the incident wave-front ; and
suppose that in unit time the disturbance from B reaches the
interface at T. In this unit-interval of time the disturbance
from A will have spread out within the crystal into a sphere
and spheroid : so the wave-front corresponding to the
ordinary ray will be the tangent-plane to the sphere through
the line whose trace is T, while the wave-front corresponding
to the extraordinary ray will be the tangent-plane to the
spheroid through the same line. The points of contact N
in the Seventeenth Century. 27
and M will determine the directions AN and A M of the two-
refracted rays* within the crystal.
Huygens did not in the Thtoi-ie de la lumiere attempt a detailed
physical explanation of the spheroidal wave, but communicated
one later in a letter to Papin,f written in December, 1690. " As
to the kinds of matter contained in Iceland crystal," he says,
" I suppose one composed of small spheroids, and another which
occupies the interspaces around these spheroids, and which serves
to bind them together. Besides these, there is the matter of
aether permeating all the crystal, both between and within the
parcels of the two kinds of matter just mentioned ; for I suppose
both the little spheroids, and the matter which occupies the
intervals around them, to be composed of small fixed particles,
amongst which are diffused in perpetual motion the still finer
particles of the aether. There is now no reason why the
ordinary ray in the crystal should not be due to waves propa-
gated in this aethereal matter. To account for the extraordinary
refraction, I conceive another kind of waves, which have for
vehicle both the aethereal matter and the two other kinds of
matter constituting the crystal. Of these latter, I suppose that
the matter of the small spheroids transmits the waves a little
more quickly than the aethereal matter, while that around the
spheroids transmits these waves a little more slowly than the
same aethereal matter. . . . These same waves, when they travel
in the direction of the breadth of the spheroids, meet with
more of the matter of the spheroids, or at least pass with less
obstruction, and so are propagated a little more quickly in this
sense than in the other ; thus the light-disturbance is propagated
as a spheroidal sheet."
Huygens made another disco veryj of capital importance when
* The word ray in the wave-theory is always applied to the line which goes
from the centre of a wave (i.e. the origin of the disturbnnce) to a point on its
surface, whatever may be the inclination of this line to the surface-element on
which it abuts; for this line has the optical properties of the "rays" of the
emission theory.
t Huygens' (Envres, ed. 1905, x., p. 177.
+ T/ieorie de la lumiere, p. 89.
28 Theory of the Aether in the Seventeenth Century.
experimenting with the Iceland crystal. He observed that the
two rays which are obtained by the double refraction of a single
ray afterwards behave in a way different from ordinary light
which has not experienced double refraction ; and in particular,
if one of these rays is incident on a second crystal of Iceland
spar, it gives rise in some circumstances to two, and in others
to only one, refracted ray. The behaviour of the ray at this
second refraction can be altered by simply rotating the second
crystal about the direction of the ray as axis ; the ray under-
going the ordinary or extraordinary refraction according as the
principal section of the crystal is in a certain direction or in the
direction at right angles to this.
The first stage in the explanation of Huygens' observation
was reached by Newton, who in 1717 showed* that a ray
obtained by double refraction differs from a ray of ordinary
light in the same way that a long rod whose cross-section is a
rectangle differs from a long rod whose cross-section is a circle :
in other words, the properties of a ray of ordinary light are the
same with respect to all directions at right angles to its direction
of propagation, whereas a ray obtained by double refraction
must be supposed to have sides, or properties related to special
directions at right angles to its own direction. The refraction
of such a ray at the surface of a crystal depends on the relation
of its sides to the principal plane of the crystal.
That a ray of light should possess such properties seemed to
Newton f an insuperable objection to the hypothesis which
regarded waves of light as analogous to waves of sound. On
this point he was in the right : his objections are perfectly
valid against the wave-theory as it was understood by his
contemporaries J, although not against the theory § which was put
forward a century later by Young and Fresnel.
* The second edition of Newton's Opticks, Query 26. t Opticks, Query 28.
J In which the oscillations are performed in the direction in which the wave
advances.
§ In which the oscillations are performed in a direction at right angles to that
in which the wave advances.
29 )
CHAPTEE II.
ELECTRIC AND MAGNETIC SCIENCE PRIOR TO THE INTRODUCTION
OF THE POTENTIALS.
THE magnetic discoveries of Peregrinus and Gilbert, and the
vortex-hypothesis by which Descartes had attempted to explain
them,* had raised magnetism to the rank of a separate science
by the middle of the seventeenth century. The kindred science
of electricity was at that time in a less developed state ; but it
had been considerably advanced by Gilbert, whose researches in
this direction will now be noticed.
For two thousand years the attractive power of amber had
been regarded as a virtue peculiar to that substance, or possessed
by at most one or two others. Gilbert provedf this view to be
mistaken, showing that the same effects are induced by friction
in quite a large class of bodies ; among which he mentioned
glass, sulphur, sealing-wax, and various precious stones.
A force which was manifested by so many different kinds of
matter seemed to need a name of its own; and accordingly
Gilbert gave to it the name electric, which it has ever since
retained.
Between the magnetic and electric forces Gilbert remarked
many distinctions. The lodestone requires no stimulus of friction
such as is needed to stir glass and sulphur into activity.
The lodestone attracts only magnetizable substances, whereas
electrified bodies attract everything. The magnetic attraction
between two bodies is not affected by interposing a sheet of
paper, or a linen cloth, or by immersing the bodies in water j
whereas the electric attraction is readily destroyed by screens.
Lastly, the magnetic force tends to arrange bodies in definite
*Cf. pp. 7-9. t De Magnete, lib. ii., cap. 2.
30 Electric and Magnetic Science
orientations ; while the electric force merely tends to heap them
together in shapeless clusters.
These facts appeared to Gilbert to indicate that electric
phenomena are due to something of a material nature, which
under the influence of friction is liberated from the glass or
amber in which under ordinary circumstances it is imprisoned.
In support of this view he adduced evidence from other quarters.
Being a physician, he was well acquainted with the doctrine
that the human body contains various humours or kinds of
moisture — phlegm, blood, choler, and melancholy, — which, as
they predominated, were supposed to determine the temper of
mind; and when he observed that electrifiable bodies were
almost all hard and transparent, and therefore (according to the
ideas of that time) formed by the consolidation of watery liquids,
he concluded that the common menstruum of these liquids must
be a particular kind of humour, to the possession of which the
electrical properties of bodies were to be referred. Friction
might be supposed to warm or otherwise excite or liberate the
humour, which would then issue from the body as an effluvium
and form an atmosphere around it. The effluvium must, he
remarked, be very attenuated, for its emission cannot be detected
by the senses.
The existence of an atmosphere of effluvia round every
electrified body might indeed have been inferred, according to
Gilbert's ideas, from the single fact of electric attraction. For
he believed that matter cannot act where it is not ; and hence
if a body acts on all surrounding objects without appearing to
touch them, something must have proceeded out of it unseen.
The whole phenomenon appeared to him to be analogous to
the attraction which is exercised by the earth on falling bodies.
For in the latter case he conceived of the atmospheric air as the
effluvium by which the earth draws all things downwards to
itself.
Gilbert's theory of electrical emanations commended itself
generally to such of the natural philosophers of the seventeenth
century as were interested in the subject ; among whom were
prior to the Introduction of the Potentials. 31
numbered Niccolo Cabeo (b. 1585, d. 1650), an Italian Jesuit
who was. perhaps the first to observe that electrified bodies repel
as well as attract ; the English royalist exile, Sir Kenelm
Digby (b. 1603, d. 1665); and the celebrated Robert Boyle
(b. 1627, d. 1691). There were, however, some differences of
opinion as to the manner in which the effluvia acted on the small
bodies and set them in motion towards the excited electric;
Gilbert himself had supposed the emanations to have an inherent
tendency to reunion with the parent body ; Digby likened their
return to the condensation of a vapour by cooling ; and other
writers pictured the effluvia as forming vortices round the
attracted bodies in the Cartesian fashion.
There is a well-known allusion to Gilbert's hypothesis in
Newton's Opticks.*
" Let him also tell me, how an electrick body can by friction
emit an exhalation so rare and subtle,t and yet so potent, as by
its emission to cause no sensible diminution of the weight of the
electrick body, and to be expanded through a sphere, whose
diameter is above two feet, and yet to be able to agitate and
carry up leaf copper, or leaf gold, at a distance of above a foot
from the electrick body ? "
It is, perhaps, somewhat surprising that the Newtonian
doctrine of gravitation should not have proved a severe blow to
the emanation theory of electricity ; but Gilbert's doctrine was
now so firmly established as to be unshaken by the overthrow
of the analogy by which it had been originally justified. It was,
however, modified in one particular about the beginning of the
eighteenth century. In order to account for the fact that
electrics are not perceptibly wasted away by excitement, the
earlier writers had supposed all the emanations to return
ultimately to the body which had emitted them ; but the
corpuscular theory of light accustomed philosophers to the
idea of emissions so subtle as to cause no perceptible loss ; and
* Query 22.
t " Subtlety," says Johnson, " which in its original import means exility of
particles, is taken in its metaphorical meaning for nicety of distinction."
32 Electric and Magnetic Science
after the time of Newton the doctrine of the return of the-
electric effluvia gradually lost credit.
Newton died in 1727. Of the expositions of his philosophy
which were published in his lifetime by his followers, one at
least deserves to be noticed for the sake of the insight which
it affords into the state of opinion regarding light, heat, and
electricity in the first half of the eighteenth century. This was
the Physices elementa matlwmatica experimentis confirmata of
Wilhelm Jacob s'Gravesande (b. 1688, d. 1742), published at
Ley den in 1720. The Latin edition was afterwards reprinted
several times, and was, moreover, translated into French and
English : it seems to have exercised a considerable and, on the
whole, well-deserved influence on contemporary thought.
s'Gravesande supposed light to consist in the projection of
corpuscles from luminous bodies to [the eye ; the motion being
very swift, as is shown by astronomical observations. Since
many bodies, e.g. the metals, become luminous when they- -are
heated, he inferred that every substance possesses a natural
store of corpuscles, which are expelled when it is heated to
incandescence ; conversely, corpuscles may become united to a
material body ; as happens, for instance, when the body is exposed
to the rays of a fire. Moreover, since the heat thus acquired is
readily conducted throughout the substance of the body, he
concluded that corpuscles can penetrate all substances, however
hard and dense they be.
Let us here recall the ideas then current regarding the
nature of material bodies. From the time of Boyle (1626-1691)
it had been recognized generally that substances perceptible to
the senses may be either elements or compounds or mixtures ;
the compounds being chemical individuals, distinct from mere
mixtures of elements. But the substances at that time accepted
as elements were very different from those which are now known
by the name. Air and the calces* of the metals figured in the
list, while almost all the chemical elements now recognized were
prior to the Introduction oj the Potentials. 33
omitted from it ; some of them, such as oxygen and hydrogen,
because they were as yet undiscovered, and others, such as the
metals, because they were believed to be compounds.
Among the chemical elements, it became customary after
the time of Newton to include light-corpuscles.* That some-
thing which is confessedly imponderable should ever have been
admitted into this class may at first sight seem surprising. But
it must be remembered that questions of ponderability counted
for very little with the philosophers of the period. Three-
quarters of the eighteenth century had passed before Lavoisier
enunciated the fundamental doctrine that the total weight of
the substances concerned in a chemical reaction is the same
after the reaction as before it. As soon as this principle came
to be universally applied, light parted company from the true
elements in the scheme of chemistry.
We must now consider the views which were held at this
time regarding the nature of heat. These are of interest for our
present purpose, on account of the analogies which were set up
between heat and electricity.
The various conceptions which have been entertained
concerning heat fall into one or other of two classes, according as
heat is represented as a mere condition producible in bodies, or
as a distinct species of matter. The former view, which is that
universally held at the present day, was advocated by the great
philosophers of the seventeenth century. Bacon maintained it in
the Novum Organum : " Calor," he wrote, " est niotus expansivus,
cohibitus, et nitens per partes minores."f Boyle+ affirmed that
the " Nature of Heat " consists in " a various, vehement, and
intestine commotion of the Parts among themselves." Hooke§
declared that " Heat is a property of a body arising from the
motion or agitation of its parts." And Newton|| asked : " Do not
* Newton himself (Oplicks, p. 349) suspected that light-corpuscles and
ponderable matter might be transmuted into each other : much later, Boscovich
(Theoria, pp. 215, 217) regarded the matter of light as a principle or element in
the constitution of natural bodies.
t Nov. Org., Lib. n., Aphor. xx. J Mechanical Production of Heat and Cold.
§ Micrographia, p. 37. || Opticks.
D
34 Electric and Magnetic Science
all fixed Bodies, when heated beyond a certain Degree, emit
light and shine ; and is not this Emission performed by the
vibrating Motion of their Parts ? " and, moreover, suggested the
converse of this, namely, that when light is absorbed by a
material body, vibrations are set up which are perceived by the
senses as heat.
The doctrine that heat is a material substance was main-
tained in Newton's lifetime by a certain school of chemists. The
most conspicuous member of the school was Wilhelm Homberg
(b. 1652, d. 1715) of Paris, who* identified heat and light with the
sulphureous principle, which he supposed to be one of the primary
ingredients of all bodies, and to be present even in the inter-
planetary spaces. Between this view and that of Newton it
might at first seem as if nothing but sharp opposition was to be
expected, j- But a few years later the professed exponents of the
Principia and the Opticks began to develop their system under
the evident influence of Homberg's writings. This evolution
may easily be traced in s'Gravesande, whose starting-point is
the admittedly Newtonian idea that heat bears to light a
relation similar to that which a state of turmoil bears to regular
rectilinear motion ; whence, conceiving light as a projection of
corpuscles, he infers that in a hot body the material particles
and the light-corpusclesj are in a state of agitation, which
becomes more violent as the body is more intensely heated.
s'Gravesande thus holds a position between the two opposite
camps. On the one hand he interprets heat as a mode of
motion ; but on the other he associates it with the presence of
a particular kind of matter, which he further identifies with the
matter of light. After this the materialistic hypothesis made
* Mem. del'Acad., 1705, p. 88.
t Though it reminds us of a curious conjecture ofNewtoa'i: "Is not the
strength and vigour of the action between light and sulphureous bodies one reason
M-liy sulphureous bodies take fire more readily and burn more vehemently than
other bodies do? "
J I have thought it best to translate s'Gravesande's ignis by " light-corpuscles."
This is, I think, fully justified by such of his statements as Quando ignis per
lineas rectas oculos nostros intrat, ex motu gttein fibris in fundo oculi cont/tninicai
ideam luminis excitat.
prior to the Introduction of the Potentials. 35
rapid progress. It was frankly advocated by another member
of the Dutch school, Hermann Boerhaave* (6. 1668, d. 1738),
Professor in the University of Leyden, whose treatise on
chemistry was translated into English in 1727.
Somewhat later it was found that the heating effects of the
rays from incandescent bodies may be separated from their
luminous effects by passing the rays through a plate of glass,
which transmits the light, but absorbs the heat. After this
discovery it was no longer possible to identify the matter of heat
with the corpuscles of light ; and the former was consequently
accepted as a distinct element, under the name of caloric.^ In
the latter part of the eighteenth and early part of the nineteenth
centuries} caloric was generally conceived as occupying the
interstices between the particles of ponderable matter — an idea
which fitted in well with the observation that bodies commonly
expand when they are absorbing heat, but which was less com-
petent to explain the fact§ that water expands when freezing.
The latter difficulty was overcome by supposing the union
between a body and the caloric absorbed in the process of
melting to be of a chemical nature; so that the consequent
changes in volume would be beyond the possibility of prediction.
As we have already remarked, the imponderability of heat
did not appear to the philosophers of the eighteenth century to
be a sufficient reason for excluding it from the list of chemical
elements ; and in any case there was considerable doubt as to
whether caloric was ponderable or not. Some experimenters
believed that bodies were heavier when cold than when hot;
others that they were heavier when hot than when cold. The
century was far advanced before Lavoisier and Eumford finally
* Boerhaave followed Homberg in supposing the matter of heat to be present ia
all so-called vacuous spaces.
t Scheele in 1777 supposed caloric to be a compound of oxygen and phlogiston,
and light to be oxygen combined with a greater proportion of phlogiston.
J In suite of the experiments of Benjamin Thompson, Count Eumford (b. 1753,
.d. 1814), in the closing years of the eighteenth century. These should have
-sufficed to re-establish the older conception of heat.
§ This had been known since the time of Boyle.
D 2
36 Electric and Magnetic Science
proved that the temperature of a body is without sensible
influence on its weight.
Perhaps nothing in the history of natural philosophy is more
amazing than the vicissitudes of the theory of heat. The true
hypothesis, after having met with general acceptance throughout
a century, and having been approved by a succession of illus-
trious men, was deliberately abandoned by their successors
in favour of a conception utterly false, and, in some of its
developments, grotesque and absurd.
We must now return to s'Gravesande's book. The pheno-
mena of combustion he explained by assuming that when a body
is sufficiently heated the light-corpuscles interact with the
material particles, some constituents being in consequence sepa-
rated and carried away with the corpuscles as flame and smoke.
This view harmonizes with the theory of calcination which had
been developed by Becher and his pupil Stahl at the end of the-
seventeenth century, according to which the metals were sup-
posed to be composed of their calces and an element phlogiston.
The process of combustion, by which a metal is changed into its-
calx, was interpreted as a decomposition, in which the phlogiston
separated from the metal and escaped into the atmosphere ;
while the conversion of the calx into the metal was regarded as
a union with phlogiston.*
s'Gravesande attributed electric effects to vibrations induced
in effluvia, which he supposed to be permanently attached to
such bodies as amber. " Glass," he asserted, " contains in it, and
has about its surface, a certain atmosphere, which is excited by
Friction and put into a vibratory motion ; for it attracts and
* The correct idea of combustion had been advanced by Hooke. "The disso-
lution of inflammable bodies," he asserts in the Micrographia, " is performed by a
substance inherent in and mixed with the air, that is like, if not the very same
with, that which is fixed in saltpetre." But this statement met with little favour
at the time, and the doctrine of the compound nature of metals survived in full
vigour until the discovery of oxygen by Priestley and Scheele in 1771-5. In 1775
Lavoisier reaffirmed Hooke's principle that a metallic calx is not the metal minus
phlogiston, but the metal plus oxygen; and this idea, which carried with it the
recognition of the elementary nature of metals, was generally accepted by the end'
of the eighteenth century.
prior to the Introduction of the Potentials. 37
repels light Bodies. The smallest parts of the glass are agitated
by the Attrition, and by reason of their elasticity, their motion is
vibratory, which is communicated to the Atmosphere above-
mentioned : and therefore that Atmosphere exerts its action the
further, the greater agitation the Parts of the Glass receive when
a greater attrition is given to the glass."
The English translator of s'Gravesande's work was himself
destined to play a considerable part in the history of electrical
science. Jean Theophile Desaguliers (b. 1683, d. 1744) was an
Englishman only by adoption. His father had been a Huguenot
pastor, who, escaping from France after the revocation of the Edict
of Nantes, brought away the boy from La Kochelle, concealed, it is
said, in a tub. The young Desaguliers was afterwards ordained,
and became chaplain to that Duke of Chandos who was so
ungratefully ridiculed by Pope. In this situation he formed
friendships with some of the natural philosophers of the capital,
and amongst others with Stephen Gray, an experimenter of
whom little is known* beyond the fact that he was a pensioner
of the Charterhouse.
In 1729 Gray communicated, as he says,f " to Dr. Desaguliers
and some other Gentlemen " a discovery he had lately made,
" showing that the Electrick Vertue of a Glass Tube may be
•conveyed to any other Bodies so as to give them the same
Property of attracting and repelling light Bodies as the Tube
does, when excited by rubbing : and that this attractive Vertue
might be carried to Bodies that were many Feet distant from
the Tube."
This was a result of the greatest importance, for previous
workers had known of no other way of producing the attractive
emanations than by rubbing the body concerned.* It was found
* Those M*ho are interested in the literary history of the eighteenth century will
recall the controversy as to whether the verses on the death of Stephen Gray were
written hy Anna "Williams, whose name they bore, or by her patron Johnson.
| Phil. Trans, xxxvii (1731), pp. 18, 227, 285, 397.
j Otto von Guericke (b. 1602, d. 1686) bad, as a matter of fact, observed the
conduction of electricity along a linen thread ; but this experiment does not seem
to have been followed up. Cf. Experimenta novamagdeburgica, 1672.
38 Electric and Magnetic Science
o
that only a limited class of substances, among which the metals
were conspicuous, had the capacity of acting as channels for the
transport of the electric power ; to these Desaguliers, who. con-
tinued the experiments after Gray's death in 1736, gavfc^ the
name non-electrics or conductors.
After Gray's discovery it was no longer possible to believe
that the electric effluvia are inseparably connected with the
bodies from which they are evoked by rubbing ; and it became
necessary to admit that these emanations have an independent
existence, and can be transferred from one body to another.
Accordingly we find them recognized, under the name of the
electric fluidft as one of the substances of which the world is
constituted. The imponderability of this fluid did not, for the
reasons already mentioned, prevent its admission by the side of
light and caloric into the list of chemical elements.
The question was actively debated as to whether the electric
fluid was an element sui generis, or, as some suspected, was
another manifestation of that principle whose operation is seen
in the phenomena of heat. Those who held the latter view
urged that the electric fluid and heat can both be induced by
friction, can both induce combustion, and can both be transferred
from one body to another by mere contact ; and, moreover, that
the best conductors of heat are also in general the best con-
ductors of electricity. On the other hand it was contended that
the electrification of a body does not cause any appreciable rise
in its temperature; and an experiment of Stephen Gray's
brought to light a yet more striking difference. Gray,J in 1729,.
made two oaken cubes, one solid and the other hollow, and
showed that when electrified in the same way they produced
exactly similar effects ; whence he concluded that it was only
the surfaces which had taken part in the phenomena. Thus
while heat is disseminated throughout the substance of a body,
the electric fluid resides at or near its surface. In the middle of
* Phil. Trans, xli. (1739), pp. 186, 193, 200, 209: Dissertation concerning
Electricity, 1742.
t The Cartesians defined a fluid to be a body whose minute parts are in a
continual agitation. J Phil. Trans, xxxvii., p. 35.
prior to the Introduction of the Potentials. 39
the eighteenth century it was generally compared to an envelop-
ing atmosphere. " The electricity which a non-electric of great
length (for example, a hempen string 800 or 900 feet long)
receives, runs from one end to the other in a sphere of electrical
Effluvia" says Desaguliers in 1740 ^and a report of the French
Academy in 1733 says :f " Around an electrified body there is
formed a vortex of exceedingly fine matter in a state of agitation,,
which urges towards the body such light substances as lie
within its sphere of activity. The existence of this vortex is
more than a mere conjecture ; for when an electrified body i&
brought close to the face it causes a sensation like that of
encountering a cobweb. "J
The report from which this is quoted was prepared in
connexion with the discoveries of Charles-Francois du Fay
(b. 1698, d. 1739), superintendent of gardens to the King of
France. Du Fay§ accounted for the behaviour of gold leaf when
brought near to an electrified glass tube by supposing that at
first the vortex of the tube envelopes the gold-leaf, and so attracts
it towards the tube. But when contact occurs, the gold-leaf
acquires the electric virtue, and so becomes surrounded by a
vortex of its own. The two vortices, striving to extend in
contrary senses, repel each other, and the vortex of the tube,
being the stronger, drives away that of the gold-leaf. " It is
then certain/' says du Fay,H " that bodies which have become
electric by contact are repelled by those which have rendered
them electric ; but are they repelled likewise by other electrified
bodies of all kinds ? And do electrified bodies differ from each
other in no respect save their intensity of electrification ? An
examination of this matter has led me to a discovery which I
should never have foreseen, and of which I believe no one
hitherto has had the least idea."
* Phil. Trans, xli., p. 636. t Hist, de 1'Acad., 1733, p. 6.
t This observation had been made first by Hawksbee at the beginning of the
century.
§ Mem. de 1'Acad. des Sciences, 1733, pp. 23, 73, 233, 457 ; 1734, pp. 341,
503; 1737, p. 86 ; Phil. Trans, xxxviii. (1734), p. 258.
|| Mem. de 1'Acad., 1733, p. 464.
40 Electric and Magnetic Science
He found, in fact, that when gold-leaf which had been
electrified by contact with excited glass was brought near to an
excited piece of copal,* an attraction was manifested between
them. " I had expected," he writes, " quite the opposite effect,
since, according to my reasoning, the copal and gold-leaf, which
were both electrified, should have repelled each other."
Proceeding with his experiments he found that the gold-leaf,
when electrified and repelled by glass, was attracted by all
electrified resinous substances, and that when repelled by the
latter it was attracted by the glass. " We see, then," he continues,
" that there are two electricities of a totally different nature —
namely, that of transparent solids, such as glass, crystal, &c.,
and that of bituminous or resinous bodies, such as amber, copal,
sealing-wax, &c. Each of them repels bodies which have
contracted an electricity of the same nature as its own, and
attracts those whose electricity is of the contrary nature. We
see even that bodies which are not themselves electrics can
acquire either of these electricities, and that then their effects
are similar to those of the bodies which have communicated it
to them."
To the two kinds of electricity whose existence was thus
demonstrated, du Fay gave the names vitreous and resinous, by
which they have ever since been known.
An interest in electrical experiments seems to have spread
from du Fay to other members of the Court circle of Louis XV ;
and from 1745 onwards the Memoirs of the Academy contain a
series of papers on the subject by the Abbe Jean-Antoine Nollet
{&. 1700, d. 1770), afterwards preceptor in natural philosophy
to the Koyal Family. Nollet attributed electric phenomena to
the movement in opposite directions of two currents of a fluid,
" very subtle and inflammable," which he supposed to be present
in all bodies under all circumstances.f When an electric is
excited by friction, part of this fluid escapes from its pores,
forming an effluent stream; and this loss is repaired by an
* A hard transparent resin, used in the preparation of varnish.
t Cf. Nollet' s lieeherchet, 1749, p. 245.
prior to the Introduction of the Potentials. 41
dtfiucnt stream of the same fluid entering the body from outside.
Light bodies in the vicinity, being caught in one or other of
these streams, are attracted or repelled from the excited electric.
Nollet's theory was in great vogue for some time ; but six or
seven years after its first publication, its author came across a
work purporting to be a French translation of a book printed
originally in England, describing experiments said to have been
made at Philadelphia, in America, by one Benjamin Franklin.
"He could not at first believe," as Franklin tells us in his
AutobiograpJvy, " that such a work came from America, and said
it must have been fabricated by his enemies at Paris to decry
his system. Afterwards, having been assured that there really
existed such a person as Franklin at Philadelphia, which he had
doubted, he wrote and published a volume of letters, chiefly
addressed to me, defending his theory, and denying the verity
of my experiments, and of the positions deduced from them."
We must now trace the events which led up to the discovery
which so perturbed Nollet.
In 1745 Pieter van Musschenbroek (6. 1692, d. 1761),
Professor at Leyden, attempted to find a method of preserving
electric charges from the decay which was observed when the
charged bodies were surrounded by air. With this purpose he
tried the effect of surrounding a charged mass of water by an
envelope of some non-conductor, e.g., glass. In one of his
experiments, a phial of water was suspended from a gun-
barrel by a wire let down a few inches into the water through
the cork; and the gun-barrel, suspended on silk lines, was
applied so near an excited glass globe that some metallic fringes
inserted into the gun-barrel touched the globe in motion.
Under these circumstances a friend named Cimaeus, who
happened to grasp the phial with one hand, and touch the gun-
barrel with the other, received a violent shock ; and it became
evident that a method of accumulating or intensifying the
electric power had been discovered.*
* The discovery was made independently in the same year by Ewald Georg
von Kleist, Dean of Kumrain.
42 Electric and Magnetic Science
o
Shortly after the discovery of the Leyden phial, as it was
named by Nollet, had become known in England, a London
apothecary named William Watson (6. 1715, d. 1787)* noticed
that when the experiment is performed in this fashion the
observer feels the shock " in no other parts of his body but his
arms and breast " ; whence he inferred that in the act of
discharge there is a transference of something which takes the
shortest or best- conducting path between the gun-barrel and
the phial. This idea of transference seemed to him to bear
some similarity to Nollet's doctrine of afflux and efflux; and
there can indeed be little doubt that the Abbe's hypothesis,
though totally false in itself, furnished some of the ideas from
which Watson, with the guidance of experiment, constructed
a correct theory. In a memoiirt)read to the Eoyal Society
in October, 1746, he propounded the doctrine that electrical
actions are due to the presence of an " electrical aether/' which
in the charging or discharging of a Leyden jar is transferred, but
is not created or destroyed. The excitation of an electric,
according to this view, consists not in the evoking of anything
from within the electric itself without compensation, but in the
accumulation of a surplus of electrical aether by the electric at
the expense of some other body, whose stock is accordingly
depleted. All bodies were supposed to possess a certain natural
store, which could be drawn upon for this purpose.
" I have shewn," wrote Watson, " that electricity is the
effect of a very subtil and elastic fluid, occupying all bodies in
contact with the terraqueous globe ; and that every-where, in
its natural state, it is of the same degree of density ; and that
glass and other bodies, which we denominate electrics per sey.
have the power, by certain known operations, of taking this fluid
from one body, and conveying it to another, in a quantity
sufficient to be obvious to all our senses; and that, under
* Watson afterwards rose to eminence in the medical profession, and was
knighted.
t Phil. Trans, xliv., p. 718. It may here he noted that it was Watson who
improved the phial by coating it nearly to the top, both inside and outside, with
tinfoil.
prior to the Introduction of the Potentials. 43
certain circumstances, it was possible to render the electricity in
some bodies more rare than it naturally is, and, by communi-
cating this to other bodies, to give them an additional quantity,
and make their electricity more dense."
In the same year in which Watson's theory was proposed, a
certain Dr. Spence, who had lately arrived in America from
Scotland, was showing in Boston some electrical experiments.
Among his audience was a man who already at forty years of
age was recognized as one of the leading citizens of the English
colonies in America, Benjamin Franklin of Philadelphia (b. 1706,
d. 1790). Spence's experiments " were," writes Franklin,*
" imperfectly performed, as he was not very expert ; but, being
on a subject quite new to me, they equally surprised and
pleased me." Soon after this, the "Library Company" of
Philadelphia (an institution founded by Franklin himself)
received from Mr. Peter Collinson of London a present of a glass
tube, with some account of its use. In a letter written to
Collinson on July llth, 1747,f Franklin described experiments
made with this tube, and certain deductions which he had
drawn from them.
If one person A, standing on wax so that electricity cannot
pass from him to the ground, rubs the tube, and if another
person B, likewise standing on wax, passes his knuckle along
near the glass so as to receive its electricity, then both A and B
will be capable of giving a spark to a third person C standing
on the floor; that is, they will be electrified. If, however, A
and B touch each other, either during or after the rubbing, they
will not be electrified.
This observation suggested to Franklin the same hypothesis
that (unknown to him) had been propounded a few months
previously by Watson : namely, that electricity is an element
present in a certain proportion in all matter in its normal
condition ; so that, before the rubbing, each of the persons A,
B, and C has an equal share. The effect of the rubbing is to
* Franklin's Autobiography.
t Franklin's New Experiments and Observations on Electricity, letter ii.
44 Electric and Magnetic Science
transfer some of A's electricity to the glass, whence it is
transferred to B. Thus A has a deficiency and B a superfluity
of electricity ; and if either of them approaches C, who has the
normal amount, the distribution will be equalized by a spark.
If, however, A and B are in contact, electricity flows between
them so as to re-establish the original equality, and neither is
then electrified with reference to C.
Thus electricity is not created by rubbing the glass, but
only transferred to the glass from the rubber, so that the
rubber loses exactly as much as the glass gains ; the, total
quantity of electricity in any insulated system is invariable. This
assertion is usually known as the principle of conservation of
electric charge.
The condition of A and B in the experiment can evidently
be expressed by plus and minus signs : A having a deficiency
- e and B a superfluity + e of electricity. Franklin, at the
commencement of his own experiments, was not acquainted
with du Fay's discoveries ; but it is evident that the electric
fluid of Franklin is identical with the vitreous electricity of
du Fay, and that du Fay's resinous electricity is, in Franklin's
theory, merely the deficiency of a stock of vitreous electricity
supposed to be possessed naturally by all ponderable bodies.
In Franklin's theory we are spared the necessity for admitting
that two quasi-material bodies can by their union annihilate each
other, as vitreous and resinous electricity were supposed to do.
Some curiosity will naturally be felt as to the considerations
which induced Franklin to attribute the positive character to
vitreous rather than to resinous electricity. They seem to have
been founded on a comparison of the brush discharges from
conductors charged with the two electricities; when the
electricity was resinous, the discharge was observed to spread
over the surface of the opposite conductor " as if it flowed from
it." Again, if a Ley den jar whose inner coating is electrified
vitreously is discharged silently by a conductor, of whose pointed
ends one is near the knob and the other near the outer coating,
the point which is near the knob is seen in the dark to be illumi-
prior to the Introduction of the Potentials. 45
nated with a star or globule, while the point which is near the
outer coating is illuminated with a pencil of rays; which
suggested to Franklin that the electric fluid, going from the
inside to the outside of the jar, enters at the former point and
issues from the latter. And yet again, in some cases the flame
of a wax taper is blown away from a brass ball which is
discharging vitreous electricity, and towards one which is
discharging resinous electricity. But Franklin remarks that
the interpretation of these observations is somewhat conjectural,
and that whether vitreous or resinous electricity is the actual
electric fluid is not certainly known.
Regarding the physical nature of electricity, Franklin held
much the same ideas as his contemporaries ; he pictured it as
an elastic* fluid, consisting of " particles extremely subtile, since
it can permeate common matter, even the densest metals, with
such ease and freedom as not to receive any perceptible
resistance." He departed, however, to some extent from the
conceptions of his predecessors, who were accustomed to ascribe
all electrical repulsions to the diffusion of effluvia from the
excited electric to the body acted on ; so that the tickling
sensation which is experienced when a charged body is brought
near to the human face was attributed to a direct action of the
effluvia on the skin. This doctrine, which, as we shall see,
practically ended with Franklin, bears a suggestive resemblance
to that which nearly a century later was introduced by
Faraday ; both explained electrical phenomena without intro-
ducing action at a distance, by supposing that something which
forms an essential part of the electrified system is present at
the spot where any electric action takes place ; but in the older
theory this something was identified with the electric fluid
itself, while in the modern view it is identified with a state of
stress in the aether. In the interval between the fall of one
school and the rise of the other, the theory of action at a
distance was dominant.
The germs of the last-mentioned theory may be found in
*i.c., repulsive of its own particles.
46 Electric and Magnetic Science
Franklin's own writings. It originated in connexion with the
explanation of the Ley den jar, a matter which is discussed
in his third letter to Collinson, of date September 1st, 1747.
In charging the jar, he says, a quantity of electricity is taken
away from one side of the glass, by means of the coating
in contact with it, and an equal quantity is communi-
cated to the other side, by means of the other coating. The
glass itself he supposes to be impermeable to the electric
fluid, so that the deficiency on the one side can permanently
coexist with the redundancy on the other, so long as the two
sides are not connected with each other ; but when a con-
nexion is set up, the distribution of fluid is equalized through
the body of the experimenter, who receives a shock.
Compelled by this theory of the jar to regard glass as
impenetrable to electric effluvia, Franklin was nevertheless well
aware* that the interposition of a glass plate between an
electrified body and the objects of its attraction does not shield
the latter from the attractive influence. He was thus driven to
supposef that the surface of the glass which is nearest the
excited body is directly affected, and is able to exert an
influence through the glass on the opposite surface ; the latter
surface, which thus receives a kind of secondary or derived
excitement, is responsible for the electric effects beyond it.
This idea harmonized admirably with the phenomena of
the jar ; for it was now possible to hold that the excess of
electricity on the inner face exercises a repellent action through
the substance of the glass, and so causes a deficiency on the
outer faces by driving away the electricity from it.J
Franklin had thus arrived at what was really a theory of
action at a distance between the particles of the electric fluid ;
and this he was able to support by other experiments. " Thus,"
he writes,§ " the stream of a fountain, naturally dense and con-
tinual, when electrified, will separate and spread in the form of
a brush, every drop endeavouring to recede from every other
* New Experiments, 1750, § 28. t Hid., 1750, § 34.
J Ibid., 1750, § 32. § Letter v.
prior to the Introduction of the Potentials. 47
drop.' In order to account for the attraction between
oppositely charged bodies, in one of which there is an excess of
electricity as compared with ordinary matter, and in the other
an excess of ordinary matter as compared with electricity, he
assumed that " though the particles of electrical matter do repel
each other, they are strongly attracted by all other matter " ; so
that " common matter is as a kind of spunge to the electrical
fluid."
These repellent and attractive powers he assigned only to
the actual (vitreous) electric fluid; and when later on the
mutual repidsion of resinously electrified bodies became known
to him,* it caused him considerable perplexity.f As we shall see,
the difficulty was eventually removed by.Aepinus.
In spite of his belief in the power of electricity to act at a
distance, Franklin did not abandon the doctrine of effluvia.
"The form of the electrical atmosphere," he says,} "is that of the
body it surrounds. This shape may be rendered visible in a still
air, by raising a smoke from dry rosin dropt into a hot tea-
spoon under the electrified body, which will be attracted, and
spread itself equally on all sides, covering and concealing the
body, And this form it takes, because it is attracted by all
parts of the surface of the body, though it cannot enter the
substance already replete. Without this attraction, it would
not remain round the body, but dissipate in the air." He
observed, however, that electrical effluvia do not seem to
affect, or be affected by, the air ; since it is possible to breathe
freely in the neighbourhood of electrified bodies ; and moreover
a current of dry air does not destroy electric attractions and
repulsions. §
Kegarding the suspected identity of electricity with the
matter of heat, as to which Nollet had taken the affirmative
position, Franklin expressed no opinion. " Common fire," he
* He refers to it in his Paper read to the Royal Society, December 18, 1755.
t Cf. letters xxxvii and xxxviii, dated 1761 and 1762.
1 New Experiment* , 1750, § 15.
§ Letter vii, 1751.
48 Electric and Magnetic Science
writes,* " is in all bodies, more or less, as well as electrical fire.
Perhaps they may be different modifications of the same
element ; or they may be different elements. The latter is by
some suspected. If they are different things, yet they may and
do subsist together in the same body."
Franklin's work did not at first receive from European
philosophers the attention which it deserved ; although Watson
generously endeavoured to make the colonial writer's merits
known,f and inserted some of Franklin's letters in one of his own
papers communicated to the Eoyal Society. But an account of
Franklin's discoveries, which had been printed in England,
happened to fall into the hands of the naturalist Buffon, who was
so much impressed that he secured the issue of a French transla-
tion of the work ; and it was this publication which, as we have
seen, gave such offence to Nollet. The success of a plan proposed
by Franklin for drawing lightning from the clouds soon engaged
public attention everywhere; and in a short time the triumph
of the one-fluid theory of electricity, as the hypothesis of
Watson and Franklin is generally called, was complete. Collet,
who was obdurate, "lived to see himself the last of his sect,
except Monsieur B — of Paris, his eleve and immediate
disciple." J
The theory of effluvia was finally overthrown, and replaced
by that of action at a distance, by the labours of one of
Franklin's continental followers, Francis Ulrich Theodore
Aepinus§ (&. 1724, d. 1802). The doctrine that glass is
impermeable to electricity, which had formed the basis of
Franklin's theory of the Ley den phial, was generalized by Aepinus||
and his co-worker Johann Karl Wilcke (5. 1732, d. 1796)
into the law that all non-conductors are impermeable to the
* Letter v.
Cx_- tPhil. Trans, xlvii, p. 202. Watson agreed with Nollet in rejecting Franklin's
J theory of the impermeability of glass.
J Franklin's Autobiography.
§ This philosopher's surname had been hellenized from its original form Hoeck
to alveivos by one of his ancestors, a distinguished theologian.
|| F. V. T. Aepinus Tentamen Thcoriae Elcctricitatis et Magnetismi :
St. Petersburg, 1759.
prior to the Introduction of the Potentials. 49
electric fluid. That this applies even to air they proved by
constructing a machine analogous to the Leyden jar, in which,
however, air took the place of glass as the medium between
two oppositely charged surfaces. The success of this experi-
ment led Aepinus to deny altogether the existence of electric
effluvia surrounding charged bodies :* a position which he
regarded as strengthened by Franklin's observation, that the
electric field in the neighbourhood of an excited body is not
destroyed when the adjacent air is blown away. The electric
fluid must therefore be supposed not to extend beyond the
excited bodies themselves. The experiment of Gray, to which
we have already referred, showed that it does not penetrate
far into their substance; and thus it became necessary to
suppose that the electric fluid, in its state of rest, is con-
fined to thin layers on the surfaces of the excited bodies.
This being granted, the attractions and repulsions observed
between the bodies compel us to believe that electricity acts
at a distance across the intervening air.
Since two vitreously charged bodies repel each other, the
force between two particles of the electric fluid must (on
Franklin's one-fluid theory, which Aepinus adopted) be
repulsive : and since there is 'an attraction between oppositely
charged bodies, the force between electricity and ordinary
matter must be attractive. These assumptions had been made,
as we have seen, by Franklin; but in order to account for
the repulsion between two resinously charged bodies, Aepinus
introduced a new supposition — namely, that the particles
of ordinary matter repel each other. This, at first, startled
his contemporaries; but, as he pointed out, the "unelectrified"
matter with which we are acquainted is really matter saturated
with its natural quantity of the electric fluid, and the forces
due to the matter and fluid balance each other ; or perhaps,
as he suggested, a slight want of equality between these
forces might give, as a residual, the force of gravitation.
Assuming that the attractive and repellent forces increase as "
* This was also maint.iined about the same time by Giacomo Battista Beet-aria
of Turin (b. 1716, d. 1781;.
E
50 Electric and Magnetic Science
<v
the distance between the acting charges decreases, Aepinus
applied his theory to explain a phenomenon which had been
more or less indefinitely observed by many previous writers, and
specially studied a short time previously by John Canton*
(&. 1718, d. 1772) and by Wilckef — namely, that if a conductor
is brought into the neighbourhood of an excited body without
actually touching it, the remoter portion of the conductor
acquires an electric charge of the same kind as that of the
excited body, while the nearer portion acquires a charge of the
opposite kind. This effect, which is known as the induction of
electric charges, had been explained by Canton himself and by
Franklin} in terms of the theory of electric effluvia. Aepinus
showed that it followed naturally from the theory of action at a
distance, by taking into account the mobility of the electric fluid
in conductors ; and by discussing different cases, so far as was
possible with the means at his command, he laid the foundations
of the mathematical theory of electrostatics.
Aepinus did not succeed in determining the law according to
which the force between two electric charges varies with the
distance between them ; and the honour of having first accom-
plished this belongs to Joseph Priestley (b. 1733, d. 1804), the
discoverer of oxygen. Priestley, who was a friend of Franklin's,
had been informed by the latter that he had found cork balls to
be wholly unaffected by the electricity of a metal cup within
which they were held ; and Franklin desired Priestley to repeat
and ascertain the fact. Accordingly, on December 21st, 1766,
Priestley instituted experiments, which showed that, when a
hollow metallic vessel is electrified, there is no charge on the inner
surface (except near the opening), and no electric force in the air
inside. From this he at once drew the correct conclusion, which
was published in 1767. § " May we not infer," he says, "from
*Phil. Trans, xlviii (1753), p. 350.
t Disputatio physica experimentalis de electricitatibus contrariis : Rostock, 1757.
J In liis paper read to the Royal Society on Dec. 18th, 1755.
§ J. Priestley, The History and Present State of Electricity, with Original
Experiments ; London, 1767: page 732. That electrical attraction follows the
law of the inverse square had been suspected -by Daniel Bernoulli in 1760: Cf.
Sochi's Experiments, Ada Helvetica, iv, p. 214.
prior to the Introduction of the Potentials. 51
this experiment that the attraction of electricity is subject to
the same laws with that of gravitation, and is therefore according
to the squares of the distances ; since it is easily demonstrated
that were the earth in the form of a shell, a body in the inside
of it would not be attracted to one side more than another ? "
This brilliant inference seems to have been insufficiently
studied by the scientific men of the day ; and, indeed, its author
appears to have hesitated to claim for it the authority of a com-
plete and rigorous proof. Accordingly we find that the question
of the law of force was not regarded as finally settled for eighteen
years afterwards.*
By Franklin's law of the conservation of electric charge, and
Priestley's law of attraction between charged bodies, electricity
was raised to the position of an exact science. It is impossible
to mention the names of these two friends in such a connexion
without reflecting on the curious parallelism of their lives. In
both men there was the same combination of intellectual bold-
ness and power with moral earnestness and public spirit. Both
.of them carried on a long and tenacious struggle with the reac-
tionary influences which dominated the English Government in
.the reign of George III ; and both at last, when overpowered in
the conflict, reluctantly exchanged their native flag for that of
the United States of America. The names of both have been
held in honour by later generations, not more for their
scientific discoveries than for their services to the cause of
religious, intellectual, and political freedom.
The most celebrated electrician of Priestley's contemporaries
in London was the Hon. Henry Cavendish (b. 1731, d. 1810),
whose interest in the subject was indeed hereditary, for his
father, Lord Charles Cavendish, had assisted in Watson's experi-
ments of 1747.f In 1771 Cavendish} presented to the Koyal
Society an " Attempt to explain some of the principal phenomena
of Electricity, by means of an elastic fluid." The hypothesis j
* In 1769 Dr. John Robison (b. 1739, d. 1805), of Edinburgh, endeavoured to
determine the law of force by direct experiment, and found it to be tbat of the
inverse 2'06th power of the distance.
t Phil. Trans, xlv, p. 67 (1750). J Phil. Trans. Ixi, p. 584 (1771).
E 2
52 Electric and Magnetic Science
adopted is that of the one-fluid theory, in much the same form
as that of Aepinus. It was, as he tells us, discovered indepen-
dently, although he became acquainted with Aepinus' work
before the publication of his own paper.
In this memoir Cavendish makes no assumption regarding
the law of force between electric charges, except that it is
" inversely as some less power of the distance than the cube " ;
but he evidently inclines to believe in the law of the inverse
square. Indeed, he shows it to be " likely, that if the electric
attraction or repulsion is inversely as the square of the distance,
almost all the redundant fluid in the body will be lodged close
to the surface, and there pressed close together, and the rest of
the body will be saturated"; which approximates closely to the
discovery made four years previously by Priestley. Cavendish
did, as a matter of fact, rediscover the inverse square law shortly
afterwards; but, indifferent to fame, he neglected to communicate
to others this and much other work of importance. The value of
his researches was not realized until the middle of the nineteenth
century, when William Thomson (Lord Kelvin) found in Caven-
dish's manuscripts the correct value for the ratio of the electric
charges carried by a circular disk and a sphere of the same radius
which had been placed in metallic connexion. Thomson urged
that the papers should be published ; which came to pass* in
1879, a hundred years from the date of the great discoveries
which they enshrined. It was then seen that Cavendish had
anticipated his successors in several of the ideas which will
presently be discussed — amongst others, those of electrostatic
capacity and specific inductive capacity.
In the published memoir of 1771 Cavendish worked out the
consequences of his fundamental hypothesis more completely
than Aepinus ; and, in fact, virtually introduced the notion of
electric potential, though, in the absence of any definite assump-
tion as to the law of force, it was impossible to develop this idea
to any great extent.
* The Electrical Researches of the Hon. Henry Cavendish, edited by J. Clerk
Maxwell, 1879.
prior to the Introduction of the Potentials. 53
One of the investigations with which Cavendish occupied
himself was a comparison between the conducting powers of
different materials for electrostatic discharges. The question
had been first raised by Beccaria, who had shown* in 1753 that
when the circuit through which a discharge is passed contains
tubes of water, the shock is more powerful when the cross-section
of the tubes is increased. Cavendish went into the matter
much more thoroughly, and was able, in a memoir presented to
the Eoyal Society in 1775,f to say : " It appears from some
experiments, of which I propose shortly to lay an account before
this Society, that iron wire conducts about 400 million times
better than rain or distilled water — that is, the electricity meets
with no more resistance in passing through a piece of iron wire
400,000,000 inches long than through a column of water of the
same diameter only one inch long. Sea- water, or a solution of
one part of salt in 30 of water, conducts 100 times, or a saturated
solution of sea-salt about 720 times, better than rain-water."
The promised account of the experiments was published in
the volume edited in 1879. It appears from it that the method
of testing by which Cavendish obtained these, results was
simply that of physiological sensation; but the figures given
in the comparison of iron and sea- water are remarkably exact.
While the theory of electricity was being established on a sure
foundation by the great investigators of the eighteenth century,
a no less remarkable development was taking place in the
kindred science of magnetism, to which our attention must now
be directed.
The law of attraction between magnets was investigated at
an earlier date than the corresponding law for electrically
charged bodies. Newton, in the Principia£ says : " The power of
gravity is of a different nature from the power of magnetism.
For the magnetic attraction is not as the matter attracted.
Some bodies are attracted more by the magnet, others less ; most
bodies not at all. The power of magnetism, in one and the same
* G. B. Beccaria, DdV ehttridsmo artificiale e natural*, Turin. 1753, p. 113.
+ Phil. Trans. Ixvi (1776), p. 196. % Book iii, Prop, vi, cor. 5.
54 Electric and Magnetic Science
body, may be increased and diminished ; and is sometimes far
stronger, for the quantity of matter, than the power of gravity ;
and in receding from the magnet, decreases not in the duplicate,
but almost in the triplicate proportion of the distance, as nearly
as I could judge from some rude observations."
The edition of ihePrincipia which was published in 1742 by
Thomas Le Seur and Francis Jacquier contains a note on this
corollary, in which the correct result is obtained that the
directive couple exercised on one magnet by another is
proportional to the inverse cube of the distance.
The first discoverer of the law of force between magnetic1
\ poles was John Michell (b. 1724, d. 1793), at that time a young
Fellow of Queen's College, Cambridge,* who in 1750 published
A Treatise of Artificial Magnets ; in ivhich is shown an easy
and expeditious method of making them superior to the lest
natural ones. In this he states the principles of magnetic
theory as followsf : —
" Wherever any Magnetism, is found, whether in the Magnet
itself, or any piece of Iron, etc., excited by the Magnet, there are
always found two Poles, which are generally called North and
South ; and the North Pole of one Magnet always attracts the
South Pole, and repels the North Pole of another: and wee versa"
This is of course adopted from Gilbert.
"Each Pole attracts or repels exactly equally, at equal
distances, in every direction." This, it may be observed, over-
throws the theory of vortices, with which it is irreconcilable.
" The Magnetical Attraction and Eepulsion are exactly equal to
each other." This, obvious though it may seem to us, was really
a most important advance, for, as he remarks, " Most people, who
* Michell had taken his degree only two years previously. Later in life he was
on terms of friendship with Priestley, Cavendish, and William Herschel ; it was
he who taught Herschel the art of grinding mirrors for telescopes. The plan of
determining the density of the earth, which was carried out by Cavendish in 1798,
and is generally known as the " Cavendish Experiment," was due to Michell.
Michell was the first inventor of the torsion-balance ; he also made many valuable
contributions to Astronomy. In 1767 he became Rector of Thornhill, Yorks,
and lived there until his death.
t Loc. cit., p. 17.
-^
prior to the Introduction of the Potentials. 55
have mention'd any thing relating to this property of the Magnet,
have agreed, not only that the Attraction and Repulsion of
Magnets are not equal to each other, but that also, they do not
observe the same rule of increase and decrease."
" The Attraction and Eepulsion of Magnets decreases, as the
Squares of the distances from the respective poles increase."
This great discovery, which is the basis of the mathematical
theory of Magnetism, was deduced partly from his own observa-
tions, and partly from those of previous investigators (e.g.
Dr. Brook Taylor and P. Muschenbroek), who, as he observes,
had made accurate experiments, but had failed to take into
account all the considerations necessary for a sound theoretical
discussion of them.
After Michell the law of the inverse square was maintained
by Tobias Mayer* of Gottingen (&. 1723, d. 1762), better known
as the author of Lunar Tables which were long in use ; and by
the celebrated mathematician, Johann Heinrich Lambertf (b.
1728, d. 1777).
The promulgation of the one-fluid theory of electricity, in
the middle of the eighteenth century, naturally led to attempts
to construct a similar theory of magnetism ; this was effected in
1759 by AepinusJ, who supposed the "poles "to be places at
which a magnetic fluid was present in amount exceeding or
falling short of the normal quantity. The permanence of
magnets was accounted for by supposing the fluid to be entangled
in their pores, so as to be with difficulty displaced. The particles
of the fluid were assumed to repel each other, and to attract the
particles of iron and steel ; but, as Aepinus saw, in order to satis-
factorily explain magnetic phenomena it was necessary to assume
also a mutual repulsion among the material particles of the
magnet.
Subsequently two imponderable magnetic fluids, to which
* Noticed in Gottinger Gelehrter Anzeiger, 1760 : cf. Aepinus, Nov. Comm.
Acad. Petrop., 1768, and Mayer's Opera Inedita, herausg. von G. C. Lichtenberg.
•\-Histoirede V Acad. de Berlin, 1766, pp. 22, 49.
% In the Tentamen, to which reference has already been made.
56 Electric and Magnetic Science
the names boreal and austral were assigned, were postulated by
the Hollander Anton Brugmans (5. 1732, d. 1789) and by
Wilcke. These fluids were supposed to have properties of
mutual attraction and repulsion similar to those possessed by
vitreous and resinous electricity.
The writer who next claims our attention for his services
both to magnetism and to electricity is the French physicist,
Charles Augustin Coulomb* (ft. 1736, d. 1806). By aid of the
torsion-balance, which was independently invented by Michell
and himself, he verified in 1785 Priestley's fundamental law
that the repulsive force between two small globes charged with
the same kind of electricity is in the inverse ratio of the square
of the distance of their centres. In the second memoir he
extended this law to the attraction of opposite electricities.
Coulomb did not accept the one-fluid theory of Franklin,
Aepinus, and Cavendish, but preferred a rival hypothesis which
had been proposed in 1759 by Kobert Symmer.f " My notion,"
said Symmer, " is that the operations of electricity do not depend
upon one single positive power, according to the opinion generally
received; but upon two distinct, positive, and active powers,
which, by contrasting, and, as it were, counteracting each other,
produce the various phenomena of electricity ; and that, when a
body is said to be positively electrified, it is not simply that it is
possessed of a larger share of electric matter than in a natural
state ; nor, when it is said to be negatively electrified, of a less ;
but that, in the former case, it is possessed of a larger portion
of one of those active powers, and in the latter, of a larger
portion of the other ; while a body in its natural state remains
unelectrified, from an equal ballance of those two powers within
it."
Coulomb developed this idea : " Whatever be the cause of
electricity," he says,J " we can explain all the phenomena by
* Coulomb's First, Second, and Third Memoirs appear in Memoires de 1'Acad.,
1785 ; the Fourth in 1786, the Fifth in 1787, the Sixth in 1788, and the Seventh
in 1789.
t Phil. Trim*, li (1759), p. 371. j Sixth Memoir, p. 561.
prior to the Introduction of the Potentials. 57
supposing that there are two electric fluids, the parts of the
same fluid repelling each other according to the inverse square
of the distance, and attracting the parts of the other fluid
according to the same inverse square law." " The supposition ^
of two fluids," he adds, " is moreover in accord with all those 7
discoveries of modern chemists and physicists, which have made
known to us various pairs of gases whose elasticity is destroyed
by their admixture in certain proportions — an effect which could
not take place without something equivalent to a repulsion
between the parts of the same gas, which is the cause of its
elasticity, and an attraction between the parts of different
gases, which accounts for the loss of elasticity on combination." J
According, then, to the two-fluid theory, the " natural fluid "
contained in all matter can be decomposed, under the influence
of an electric field, into equal quantities of vitreous and
resinous electricity, which, if the matter be conducting, can then
fly to the surface of the body. The abeyance of the characteristic
properties of the opposite electricities when in combination was f
sometimes further compared to the neutrality manifested by .
the compound of an acid and an alkali.
The publication of Coulomb's views led to some controversy
between the partisans of the one-fluid and two-fluid theories ; the
latter was soon generally adopted in France, but was stoutly
opposed in Holland by Van Marum and in Italy by Volta.
The chief difference between the rival hypotheses is that, in the ^
two-fluid theory, both the electric fluids are movable within the
substance of a solid conductor ; while in the one-fluid theory the
actual electric fluid is mobile, but the particles of the conductor
are fixed. The dispute could therefore be settled only by a deter-
mination of the actual motion of electricity in discharges ; and
this was beyond the reach of experiment.
In his Fourth Memoir Coulomb showed that electricity in
equilibrium is confined to the surface of conductors, and does
not penetrate to their interior substance ; and in the Sixth
Memoir* he virtually establishes the result that the electric
* Page 677.
58 Electric and Magnetic Science
force near a conductor is proportional to the surface-density of
electrification.
Since the overthrow of the doctrine of electric effluvia by
Aepinus, the aim of electricians had been to establish their
science upon the foundation of a law of action at a distance,
resembling that which had led to such triumphs in Celestial
Mechanics. When the law first stated by Priestley was at
length decisively established by Coulomb, its simplicity and
beauty gave rise to a general feeling of complete trust in it as
the best attainable conception of electrostatic phenomena.
The result was that attention was almost exclusively focused
on action-at-a-distance theories, until the time, long afterwards,,
when Faraday led natural philosophers back to the right'
path.
Coulomb rendered great services to magnetic theory. It was
he who in 1777, by simple mechanical reasoning, completed
the overthrow of the hypothesis of vortices.* He also, in the
second of the Memoirs already quoted,f confirmed Michell's
law, according to which the particles of the magnetic fluids
attract or repel each other with forces proportional to the
inverse square of the distance. Coulomb, however, went beyond
this, and endeavoured to account for the fact that the two
magnetic fluids, unlike the two electric fluids, cannot be
obtained separately; for when a magnet is broken into
two pieces, one containing its north and the other its south
pole, it is found that each piece is an independent magnet
possessing two poles of its own, so that it is impossible
to obtain a north or south pole in a state of isolation.
Coulomb explained this by supposing^ that the mag-
netic fluids are permanently imprisoned within the molecules
of magnetic bodies, so as to be incapable of crossing from
one molecule to the next ; each molecule therefore under all
circumstances contains as much of the boreal as of the
* Mem. presences par divers Savans, ix (1780), p. 165.
t Mem de 1'Acad., 1785, p. 593. Gauss finally established the law by a
much more refined method.
J In his Seventh Memoir, Mem, de 1'Acad., 1789, p. 488.
prior to the Introduction of the Potentials. 59
austral fluid, and magnetization consists simply in a separation
of the two fluids to opposite ends of each molecule. Such
a hypothesis evidently accounts for the impossibility of
separating the two fluids to opposite ends of a body of finite
size. The same idea, here introduced for the first time, has
since been applied with success in other departments of
electrical philosophy.
In spite of the advances which have been recounted,
the mathematical development of electric and magnetic theory
was scarcely begun at the close of the eighteenth century ; and
many erroneous notions were still widely entertained. In a
Eeport* which was presented to the French Academy in 1800,
it was assumed that the mutual repulsion of the particles of
electricity on the surface of a body is balanced by the
resistance of the surrounding air; and for long afterwards
the electric force outside a charged conductor was confused
with a supposed additional pressure in the atmosphere.
Electrostatical theory was, however, suddenly advanced to
quite a mature state of development by Simeon Denis Poisson
(b. 1781, d. 1840), in a memoir which was read to the French
Academy in 1812.f As the opening sentences show, he accepted
the conceptions of the two-fluid theory.
" The theory of electricity which is most generally accepted,"
he says, " is that which attributes the phenomena to two
different fluids, which are contained in all material bodies.
It is supposed that molecules of the same fluid repel each
other and attract the molecules of the other fluid ; these
forces of attraction and repulsion obey the law of the inverse
square of the distance ; and at the same distance the attractive
power is equal to the repellent power; whence it follows
that, when all the parts of a body contain equal quantities
of the two fluids, the latter do not exert any influence on
the fluids contained in neighbouring bodies, and consequently
no electrical effects are discernible. This equal and uniform
* On Yolla's discoveries.
t Mem. de Plnstitut, 1811, Part i., p. 1, Part ii., p. 163.
60 Electric and Magnetic Science
distribution of the two fluids is called the natural state ; when this
state is disturbed in any body, the body is said to be electrified,
and the various phenomena of electricity begin to take place.
"Material bodies do not all behave in the same way with
respect to the electric fluid : some, such as the metals, do
not appear to exert any influence on it, but permit it to
move about freely in their substance ; for this reason they
are called conductors. Others, on the contrary — very dry air,
for example — oppose the passage of the electric fluid in their
interior, so that they can prevent the fluid accumulated in
conductors from being dissipated throughout space."
When an excess of one of the electric fluids is communi-
cated to a metallic body, this charge distributes itself over the
surface of the body, forming a layer whose thickness at any
point depends on the shape of the surface. The resultant force
due to the repulsion of all the particles of this surface-layer
must vanish at any point in the interior of the conductor, since
otherwise the natural state existing there would be disturbed ;
and Poisson showed that by aid of this principle it is possible
in certain cases to determine the distribution of electricity in
the surface-layer. For example, a well-known proposition of
the theory of Attractions asserts that a hollow shell whose
bounding surfaces are two similar and similarly situated
ellipsoids exercises 110 attractive force at any point within the
interior hollow; and it may thence be inferred that, if an
electrified metallic conductor has the form of an ellipsoid, the
charge will be distributed on it proportionally to the normal
distance from the surface to an adjacent similar and similarly
situated ellipsoid.
Poisson went on to show that this result was by no means all
• that might with advantage be borrowed from the theory of
I Attractions. Lagrange, in a memoir on the motion of gravitating
bodies, had shown* that the components of the attractive force
* Mem. de Berlin, 1777. The theorem was afterwards published, and ascribed
to Laplace, in a memoir by Legendre on the Attractions of Spheroids, which will
be found in the Mem. par divers Snvanx, published in 178o.
prior to the Introduction of the Potentials. 61
at any point can be simply expressed as the derivates of the
function which is obtained by adding together the masses of all
the particles of an attracting system, each divided by its
distance from the point; and Laplace had shown* that this
function V satisfies the equation
in space free from attracting matter. Poisson himself showed
later, in 1813,f that when the point (z, y, z) is within the
substance of the attracting body, this equation of Laplace must
be replaced by
W VV VV
^ + w~~vr: p>
where p denotes the density of the attracting matter at the
point. In the present memoir Poisson called attention to the
utility of this function F in electrical investigations, remarking
that its value over the surface of any conductor must be
constant.
The known formulae for the attractions of spheroids show
that when a charged conductor is spheroidal, the repellent force
acting on a small charged body immediately outside it will be
directed at right angles to the surface of the spheroid, and will
be proportional to the thickness of the surface-layer of electricity
at this place. Poisson suspected that this theorem might be
true for conductors not having the spheroidal form — a result
which, as we have seen, had been already virtually given by
Coulomb ; and Laplace suggested to Poisson the following
proof, applicable to the general case. The force at a point
immediately outside the conductor can be divided into a
part s due to the part of the charged surface immediately
adjacent to the point, and a part S due to the rest of
the surface. At a point close to this, but just inside the con-
ductor, the force j^jpll still act; but the forces will evidently
* Mem. de 1'Acad., 1782 (published in 1785), p. 113.
t Bull, de la Soc. Philomathique. iii. (1813,, p. 388.
62 Electric and Magnetic Science
be reversed in direction. Since the resultant force at the latter
point vanishes, we must have S=s ; so the resultant force at the
exterior point is 2s. But s is proportional to the charge per
unit area of the surface, as is seen by considering the case of
an infinite plate ; which establishes the theorem.
When several conductors are in presence of each other, the
distribution of electricity on their surfaces may be determined
by the principle, which Poisson took as the basis of his work,
that at any point in the interior of any one of the conductors,
the resultant force due to all the surf ace -layers must be zero.
He discussed, in particular, one of the classical problems of
electrostatics — namely, that of determining the surface-density
on two charged conducting spheres placed at any distance from
each other. The solution depends on Double Gamma Functions
in the general case ; when the two spheres are in contact, it
depends on ordinary Gamma Functions. Poisson gave a solution
in terms of definite integrals, which is equivalent to that in
terms of Gamma Functions ; and after reducing his results to
numbers, compared them with Coulomb's experiments.
f The rapidity with which in a single memoir Poisson passed
from the barest elements of the subject to such recondite
problems as those just mentioned may well excite admiration.
His success is, no doubt, partly explained by the high state of
development to which analysis had been advanced by the great
mathematicians of the eighteenth century ; but even after
allowance has been made for what is due to his predecessors,
Poisson' s investigation must be accounted a splendid memorial
uof his genius.
Some years later Poisson turned his attention to magnetism ;
and, in a masterly paper* presented to the French Academy in
1824, gave a remarkably complete theory of the subject.
His starting-point is Coulomb's doctrine of two imponderable
magnetic fluids, arising from the decomposition of a neutral
fluid, and confined in their movements to the individual elements
* Mem. <le 1'Acad., v, p. 247.
prior to the Introduction of the Potentials. 63
of the magnetic body, so as to be incapable of passing from one
element to the next
Suppose that an amount m of the positive magnetic fluid is
located at a point (x y, z) ; the components of the magnetic
intensity, or force exerted on unit magnetic pole, at a point
(£, »f, £) will evidently be
-m-f-X -m~(-\ -m-(-)
where r denotes ((? - xf + (n - ?/)2 + (Z - z)2j*. Hence if we
consider next a magnetic element in which equal quantities of
the two magnetic fluids are displaced from each other parallel
to_ the ic-axis, the components of the magnetic intensity at
(g, i|, 2) will be the negative derivates, with respect to £ ij, £
respectively, of the function
where the quantity A, which does not involve (f, »j, £), may be
called the magnetic moment of the element : it may be measured
by the couple required to maintain the element in equilibrium
at a definite angular distance from the magnetic meridian.
If the displacement of the two fluids from each other in the
element is not parallel to the axis of xt it is easily seen that the
expression corresponding to the last is
where the vector (A, B, C) now denotes the magnetic moment
of the element.
Thus the magnetic intensity at an -external point (£, 77, £)
due to any magnetic body has the components
«; - 017
where
ex oy
integrated throughout the substance of the magnetic body, and
64 Electric and Magnetic Science
where the vector (A, B, C) or I represents the magnetic moment
per unit- volume, or, as it is generally called, the magnetization.
The function Fwas afterwards named by Green the magnetic
potential.
Poisson, by integrating by parts the preceding expression for
the magnetic potential, obtained it in the form
F = [[(I . dS). \ - fjp div I dx dy dz*
the first integral being taken over the surface $ of the magnetic
body, and the second integral being taken throughout its volume.
This formula shows that the magnetic intensity produced by the
body in external space is the same as would be produced by a
fictitious distribution of magnetic fluid, consisting of a layer
over its surface, of surface-charge (I .- dS) per element dSy
together with a volume-distribution of density - div I through-
out its substance. These fictitious magnetizations are generally
known as Poisson's equivalent surface- and volume-distributions
of magnetism.
Poisson, moreover, perceived that at a point in a very small
cavity excavated within the magnetic body, the magnetic
potential has a limiting value which is independent of the shape
of the cavity as the dimensions of the cavity tend to zero ; but
that this is not true of the magnetic intensity, which in such a
small cavity depends on the shape of the cavity. Taking the
cavity to be spherical, he showed that the magnetic intensity
within it is
grad F 4 ^-7rl,f
where I denotes the magnetization at the place.
* If the components of a vector a are denoted by (ax, ay, az), the quantity
drbjc + ayby -f- atkz is called the scalar product of two vectors a and b, and is denoted
by (a . b).
The quantity ^— ' + ^ + ^ is called the divergence of the vector a, and is
fix dy 02
denoted by div a.
t The vector whose components are - — , - •?—, - -„— is denoted by grad V.
C£ dy dz J °
prior to the Introduction of the Potentials. 65
This memoir also contains a discussion of the magnetism
temporarily induced in soft iron and other magnetizable metals
by the approach of a permanent magnet. Poisson accounted for
the properties of temporary magnets by assuming that they
contain embedded in their substance a great number of small
spheres, which are perfect conductors for the magnetic fluids ; so
that the resultant magnetic intensity in the interior of one of
these small spheres must be zero. He showed that such a sphere,
when placed in a field of magnetic intensity F,* must acquire a
magnetic moment of amount -.- F x the volume of the sphere,
in order to counteract within the sphere the force F. Thus if
kp denote the total volume of these spheres contained within a
unit volume of the temporary magnet, the magnetization will be
I, where 4-TrI = kp F,
and F denotes the magnetic intensity within a spherical cavity
excavated in the body. This is Poisson s laiv of induced magnetism.
It is known that some substances acquire a greater degree
of temporary magnetization than others when placed in the
same circumstances : Poisson accounted for this by supposing that
the quantity kp varies from one substance to another. But the
experimental data show that for soft iron kp must have a value
very near unity, which would obviously be impossible if kp is to
mean the ratio of the volume of spheres contained within a
region to the total volume of the region.f The physical inter-
pretation assigned by Poisson to his formulae must therefore be
rejected, although the formulae themselves retain their value.
Poisson's electrical and magiietical investigations were
generalized and extended in 1828 by George Green* (b. 1793,
d. 1841). Green's treatment is based on the properties of the
function already used by Lagrange, Laplace, and Poisson, which
* In the present work, vectors will generally be distinguished by heavy type.
t This objection was advanced by Maxwell in § 430 of his Treatise. An attempt
to overcome it was made by Betti : cf. p. 377 of his Lessons on the Potential.
J A.n essay on the application of mathematical analysis to the theories of electricity
and magnetism, Nottingham, 1828 : reprinted in The Mathematical Papers of the late
George Green, p. 1.
F
66 Electric and Magnetic Science.
represents the sum of all the electric or magnetic charges in the
field, divided by their respective distances from some given point :
to this function Green gave the name potential, by which it has
always since been known.*
Near the beginning of the memoir is established the
celebrated formula connecting surface and volume integrals,
which is now generally called G-reeris Theorem, and of which
Poisson's result on the equivalent surface- and volume-distribu-
tions of magnetization is a particular application. By using
this theorem to investigate the properties of the potential,
Green arrived at many results of remarkable beauty and
interest. We need only mention, as an example of the power
of his method, the following : — Suppose that there is a hollow
conducting shell, bounded by two closed surfaces, and that a
number of electrified bodies are placed, some within and some
without it ; and let the inner surface and interior bodies be
called the interior system, and the outer surface and exterior
botlies be called the exterior system. Then all the electrical
phenomena of the interior system, relative to attractions,
repulsions, and densities, will be the same as if there were no
exterior system, and the inner surface were a perfect conductor,
put in communication with the earth ; and all those of the
exterior system will be the same as if the interior system did not
exist, and the outer surface were a perfect conductor, containing
a quantity of electricity equal to the whole of that originally
contained in the shell itself and in all the interior bodies.
It will be evident that electrostatics had by this time
attained a state of development in which further progress could
be hoped for only in the mathematical superstructure, unless
experiment should unexpectedly bring to light phenomena of
an entirely new character. This will therefore be a convenient
place to pause and consider the rise of another branch of
electrical philosophy.
* Euler in 1744 (De melhodis inveniendi . . .) had spoken of the vis potentialis —
what would now be called the potential energy — possessed by an elastic body
when bent.
CHAPTEE III.
GALVANISM, FROM GALVANI TO OHM.
UNTIL the last decade of the eighteenth century, electricians
were occupied solely with statical electricity. Their attention
was then turned in a different direction.
In a work entitled Recherches sur Vorigine des sentiments
agreables et cUsagr cables, which was published* in 1752,
Johann Georg Sulzer (b. 1720, d. 1779) had mentioned that, if
two pieces of metal, the one of lead and the other of silver, be
joined together in such a manner that their edges touch, and if
they be placed on the tongue, a taste is perceived " similar to
that of vitriol of iron," although neither of these metals applied
separately gives any trace of such a taste. " It is not probable,"
he says, " that this contact of the two metals causes a solution
of either of them, liberating particles which might affect the
tongue : and we must therefore conclude that the contact sets
up a vibration in their particles, which, by affecting the nerves
of the tongue, produces the taste in question."
This observation was not suspected to have any connexion
with electrical phenomena, and it played no part in the incep-
tion of the next discovery, which indeed was suggested by a
mere accident.
Luigi Galvani, born at Bologna in 1737, occupied from 1775
onwards a chair of Anatomy in his native city. For many years
before the event which made him famous he had been studying
the susceptibility of -the nerves to irritation ; and, having been <-
formerly a pupil of Beccaria, he was also interested in electrical
experiments. One day in the latter part of the year 1780 he '
had, as he tells us,f " dissected and prepared a frog, and laid it
on a table, on which, at some distance from the frog, was an
electric machine. It happened by chance that one of my
* Mem. de 1'Acad. de Berlin, 1752, p. 356.
t Aloysii Galvani, De Viribus E 'lee trie itatis in Motu Mnsculari : Commentarii
Bononiensi, vii (1791), p. 363.
F 2
68 Galvanism, from Galvani to Ohm.
assistants touched the inner crural nerve of the frog with the
point of a scalpel ; whereupon at once the muscles of the limbs
were violently convulsed.
" Another of those who used to help me in electrical experi-
ments thought he had noticed that at this instant a spark was
drawn from the conductor of the machine. I myself was at the
time occupied with a totally different matter; but when he
drew my attention to this, I greatly desired to try it for myself,.
and discover its hidden principle. So I, too, touched one or
other of the crural nerves with the point of the scalpel, at the
same time that one of those present drew a spark ; and the same
phenomenon was repeated exactly as before."*
After this, Galvani conceived the idea of trying whether the
electricity of thunderstorms would induce muscular contractions
equally well with the electricity of the machine. Having
successfully experimented with lightning, he " wished," as he
writes,! " to try the effect of atmospheric electricity in calm
weather. My reason for this was an observation I had made,,
that frogs which had been suitably prepared for these experi-
ments and fastened, by brass hooks in the spinal marrow, to
the iron lattice round a certain hanging-garden at my house,,
exhibited convulsions not only during thunderstorms, but
sometimes even when the sky was quite serene. I suspected
these effects to be due to the changes which take place during
the day in the electric state of the atmosphere ; and so, with
some degree of confidence, I performed experiments to test the
point; and at different hours for many days I watched frogs
which I had disposed for the purpose ; but could not detect any
motion in their muscles. At length, weary of waiting in vain,
I pressed the brass hooks, which were driven into the spinal
marrow, against the iron lattice, in order to see whether
contractions could be excited by varying the incidental circum-
* According to a story which has often been repeated, but which rests on no
sufficient evidence, the frog was one of a number which had been procured for th&
Signora Galvani, who, being in poor health, had been recommended to take a soup,
made of these animals as a restorative. f Loc. cit., p. 377.
Galvanism, from Galvani to Ohm. 69
stances of the experiment. I observed contractions tolerably
often, but they did not seem to bear any relation to the changes
in the electrical state of the atmosphere.
" However, at this time, when as yet I had not tried the
experiment except in the open air, I came very near to adopt-
ing a theory that the contractions are due to atmospheric
electricity, which, having slowly entered the animal and accu-
mulated in it, is suddenly discharged when the hook comes in
contact with the iron lattice. For it is easy in experimenting
to deceive ourselves, and to imagine we see the things we wish
to see.
" But I took the animal into a closed room, and placed it on
an iron- plate ; and when I pressed the hook which was fixed
in the spinal marrow against the plate, behold ! the same
spasmodic contractions as before. I tried other metals at
different hours on various days, in several places, and always
with the same result, except that the contractions were more
violent with some metals than with others. After this I tried
various bodies which are not conductors of electricity, such as
glass, gums, resins, stones, and dry wood ; but nothing happened.
This was somewhat surprising, and led me to suspect that
electricity is inherent in the animal itself. This suspicion was
strengthened by the observation that a kind of circuit of subtle
nervous fluid (resembling the electric circuit which is manifested
in the Leyclen jar experiment) is completed from the nerves to
the muscles when the contractions are produced.
" For, while I with one hand held the prepared frog by the
hook fixed in its spinal marrow, so that it stood with its feet
on a silver box, and with the other hand touched the lid of
the box, or its sides, with any metallic body, I was surprised
to see the frog become strongly convulsed every time that I
applied this artifice."*
Galvani thus ascertained that the limbs of the frog are con-
vulsed whenever a connexion is made between the nerves and
muscles by a metallic arc, generally formed of more than one
*This observation was made in 1786.
70 Galvanism > from Galvani to Ohm.
kind of metal ; and he advanced the hypothesis that the convul-
sions are caused by the transport of a peculiar fluid from the
' nerves to the muscles, the arc acting as a conductor. To this
fluid the names Galvanism and .Animal Electricity were soon
generally applied. Galvani himself considered it to be the same
as the ordinary electric fluid, and, indeed, regarded the entire
phenomenon as similar to the discharge of a Leyden jar.
*' The publication of Gralvani's views soon engaged the attention
of the learned world, and gave rise to an animated controversy
between those who supported Galvani's own view, those who
believed galvanism to be a fluid distinct from ordinary electricity,
and a third school who altogether refused to attribute the effects
to a supposed fluid contained in the nervous system. The leader
of the last-named party was Alessandro Volta (b. 1745, d. 1827),
Professor of Natural Philosophy in the University of Pavia, who
in 1792 put forward the view* that the stimulus in Galvani's
experiment is derived essentially from the connexion of two
different metals by a moist body. "The metals used in the
* experiments, being applied to the moist bodies of animals, can by
themselves, and of their proper virtue, excite and dislodge the
electric fluid from its state of rest ; so that the organs of the
* animal act only passively." At first he inclined to combine this
theory of metallic stimulus with a certain degree of belief in
such a fluid as Galvani had supposed; but after the end of 17!. '3
he denied the existence of animal electricity altogether.
From this standpoint Volta continued his experiments and
worked out his theory. The following quotation from a lettert
which he wrote later to Gren, the editor of the Neucs Journal //.
Physik, sets forth his view in a more developed form : —
"The contact of different conductors, particularly the metallic,
including pyrites and other minerals, as well as charcoal, which
I call dry conductors, or of the first class, with moist conductors,
or conductors of the second class, agitates or disturbs the electric
f fluid, or gives it a certain impulse. Do not ask in what manner :
it is enough that it is a principle, and a general principle. This
*Phil. Trans., 1793, pp. 10, 27. tPhil. Mag. iv (1799), pp. 59, 163, 306.
Galvanism , from Galvani to Okm. 71
impulse, whether produced by attraction or any other force, is
different or unlike, both in regard to the different metals and to
the different moist conductors ; so that the direction, or at least
the power, with which the electric fluid is impelled or excited, is
different when the conductor A is applied to the conductor B, or
to another C. In a perfect circle of conductors, where either
one of the second class is placed between two different from each
other of the first class, or, contrariwise, one of the first class is
placed between two of the second class different from each other,
an electric stream is occasioned by the predominating force either
to the right or to the left — a circulation of this fluid, which ceases
only when the circle is broken, and which is renewed when the
circle is again rendered complete."
Another philosopher who, like Volta, denied the existence of
a fluid peculiar to animals, but who took a somewhat different
view of the origin of the phenomenon, was Giovanni Fabroni, of
Florence (b. 1752, d. 1822), who,* having placed two plates of
different metals in water, observed that one of them was partially
oxidized when they were put in contact ; from which he rightly
concluded that some chemical action is inseparably connected
with galvanic effects.
The feeble intensity of the phenomena of galvanism, which
compared poorly with the striking displays obtained in electro-
statics, was responsible for some falling off of interest in them
towards the end of the eighteenth century ; and the last years
of their illustrious discoverer were clouded by misfortune. Being
attached to the old order which was overthrown by the armies
of the French Ke volution, he refused in 1798 to take the oath of
allegiance to the newly constituted Cisalpine Eepublic, and was
deposed from his professorial chair. A profound melancholy,
which had been induced by domestic bereavement, was aggra-
vated by poverty and disgrace ; and, unable to survive the loss
of all he held dear, he died broken-hearted before the end of
the year.f
* Phil. Journal, 4to, iii. 308 ; iv. 120 ; Journal de Physique, vi. 348.
t A decree of reinstatement had been granted, but had not come into operation
at the time of Galvani's death.
<
72 Galvanism, Jrom Galvani to O/it/i.
Scarcely more than a year after the death of Galvani, the
new science suddenly regained ' the eager attention of philo-
sophers. This renewal of interest was due to the discovery by
Volta, in the early spring of 1800, of a means of greatly increasing
the intensity of the effects. Hitherto all attempts to magnify
the action by enlarging or multiplying the apparatus had ended
in failure. If a long chain of different metals was used instead
of only two, the convulsions of the frog were no more violent.
But Volta now showed* that if any number of couples, each
consisting of a zinc disk and a copper disk in contact, were taken,
and if each couple was separated from the next by a disk of moist-
ened pasteboard (so that the order was copper, zinc, pasteboard,
copper, zinc, pasteboard, &c.), the effect of the pile thus formed
was much greater than that of any galvanic apparatus previously
introduced. When the highest and lowest disks were simul-
taneously touched by the fingers, a distinct shock was felt ; and
this could be repeated again and again, the pile apparently
possessing within itself an indefinite power of recuperation. It
thus resembled a Leyden jar endowed with a power of automati-
cally re-establishing its state of tension after each explosion;
with, in fact, " an inexhaustible charge, a perpetual action or
impulsion on the electric fluid."
Volta unhesitatingly pronounced the phenomena of the pile
to be in their nature electrical. The circumstances of Galvani's
original discovery had prepared the minds of philosophers for
this belief, which was powerfully supported by the similarity of
the physiological effects of the pile to those of the Leyden jar,
and by the observation that the galvanic influence was conducted
only by those bodies — e.g. the metals — which were already
known to be good conductors of static electricity. But Volta
now supplied a still more convincing proof. Taking a disk of
copper and one of zinc, 'he held each by an insulating handle
and applied them to each other for an instant. After the disks
had been separated, they were brought into contact with a deli-
* I'hil. Trans., 1800, p. 403.
Galvanism, from Galvani to Ohm. 73
oate electroscope, which indicated by the divergence of its straws
that the disks were now electrified — the zinc had, in fact, acquired
a positive and the copper a negative electric charge.* Thus the
mere contact of two different metals, such as those employed in /
the pile, was shown to be sufficient for the production of effects '
undoubtedly electrical in character.
On the basis of this result Volta in the same year (1800)
put forward a definite theory of the action of the pile. Suppose
first that a disk of zinc is laid on a disk of copper, which in turn
rests on an insulating support. The experiment just described
shows that the electric fluid will be driven from the copper to
the zinc. We may then, according to Volta, represent the state
or " tension " of the copper by the number - J, and that of the
zinc by the number + J, the difference being arbitrarily taken as
unity, and the sum being (on account of the insulation) zero. It
will be seen that Volta's idea of " tension " was a mingling of
two ideas, which in modern electric theory are clearly distin-
guished from each other — namely, electric charge and electric
potential.
Now let a disk of moistened pasteboard be laid on the zinc,
and a disk of copper on this again. Since the uppermost
copper is not in contact with the zinc, the contact-action does
not take place between them ; but since the moist pasteboard is
a conductor, the copper will receive a charge from the zinc.
Thus the states will now be represented by - f for the lower
copper, + J for the zinc, and + \ for the upper copper, giving a
zero sum as before.
If, now, another zinc disk is placed on the top, the states
will be represented by - 1 for the lower copper, 0 for the lower
zinc and upper copper, and + 1 for the upper zinc.
In this way it is evident that the difference between the
numbers indicating the tensions of the uppermost and lowest
* Abraham Bennet (b. 1750, d. 1799) had previously shown (Xew Experiments
in Electricity, 1789, pp. 86-102) that many bodies, when separated after contact, f
are oppositely electrified ; he conceived that different bodies have different attrac-
tions or capacities for electricity.
74 Galvanism , from Galvani to O/im.
disks in the pile will always be equal to the number of pairs of
metallic disks contained in it. If the pile is insulated, the
sum of the numbers indicating the states of all the disks must
be zero; but if the lowest disk is connected to earth, the
tension of this disk will be zero, and the numbers indicating the
states of all the other disks will be increased by the same
amount, their mutual differences remaining unchanged.
The pile as a whole is thus similar to a Leyden jar ;
when the experimenter touches the uppermost and lowest
disks, he receives the shock of its discharge, the intensity being
proportional to the number of disks.
The moist layers played no part in Volta's theory beyond
j. that of conductors.* It was soon found that when the moisture
is acidified, the pile is more efficient; but this was attributed
solely to the superior conducting power of acids.
Yolta fully understood and explained the impossibility of
constructing a pile from disks of metal alone, without making
use of moist substances. As he showed in 1801, if disks of
various metals are placed in contact in any order, the extreme
metals will be in the same state as if they touched each other
directly without the intervention of the others ; so that the
whole is equivalent merely to a single pair. When the metals
are arranged in the order silver, copper, iron, tin, lead, zinc,
each of them becomes positive with respect to that which
precedes it, and negative with respect to that which follows it ;
but the moving force from the silver to the zinc is equal to the
sum of the moving forces of the metals comprehended between
them in the series.
When a connexion was maintained for some time between
the extreme disks of a pile by the human body, sensations
were experienced which seemed to indicate a continuous activity
in the entire system. Yolta inferred that the electric current
persists during the whole time that communication by con-
* Volta had inclined, in his earlier experiments on galvanism, to locate the seat
of power at the interfaces of the metals with the rnoist conductors. Cf. his letter
to Gren, Phil. Mag. iv (1799), p. 62.
Galvanism, from Gaivani to Ohm. 75
ductors exists all round the circuit, and that the current is
suspended only when this communication is interrupted.
" This endless circulation or perpetual motion of the electric
fluid," he says, "may seem paradoxical, and may prove
inexplicable ; but it is none the less real, and we can, so to
speak, touch and handle it."
Yolta announced his discovery in a letter to Sir Joseph
Banks, dated from Como, March 20th, 1800. Sir Joseph, who
was then President of the Eoyal Society, communicated the
news to William Nicholson (b. 1753, d. .1815), founder of the
Journal which is generally known by his name, and his
friend Anthony Carlisle (b. 1768, d. 1840), afterwards a
distinguished surgeon. On the 30th of the following month,
Nicholson and Carlisle set up the first pile made in England. In
repeating Volta's experiments, having made the contact more
secure at the upper plate of the pile by placing a drop of water
there, they noticed* a disengagement of gas round the con-
ducting wire at this point ; whereupon they followed up the
matter by introducing a tube of water, into which the wires
from the terminals of the pile were plunged. Bubbles of an
inflammable gas were liberated at one wire, while the other
wire became oxidised ; when platinum wires were used, oxygen
and hydrogen were evolved in a free state, one at each wire.
This effect, which was nothing less than the electric decom-
position of water into its constituent gases, was obtained on
May 2nd, 1800.f
Although it had long been known that frictional electricity
is capable of inducing chemical action,* the discovery of
Nicholson and Carlisle was of the first magnitude. It was at
once extended by William Cruickshank, of Woolwich (b. 1745,
i's Journal (4to), iv, 179 (1800) ; Phil. Mag. vii, 337 (1800).
t It was obtained independently four months later l>y J. "W. Hitter.
J Beccaria (Lettere deW elettricismo, Bologna, 1758, p. 282) had reduced mercury
and other metals from their oxides by discharges ot fractional electricity ; and
Priestley had obtained an inflammable gas from certain organic liquids in the
same way. Cavendish in 1781 had established the constitution of water by
electrically exploding hydrogen and oxygen.
76 Galvanism > from Galvani to Ohm.
d. 1800), who* showed that solutions of metallic salts are also
decomposed by the current; and William Hyde Wollaston
(ft. 1766, d. 1828) seized on it as a testf of the identity of the
electric currents of Volta with those obtained by the discharge
of f rictional electricity. He found that water could be decom-
vy posed by currents of either type, and inferred that all differences
between them could be explained by supposing that voltaic
electricity as commonly obtained is " less intense, but produced
in much, larger quantity." Later in the same year (1801),
Martin van Mar um (ft. 1750, d. 1837) and Christian Heinrich
Pfaff (ft. 1773, d. 1852) arrived at the same conclusion by
carrying out on a large scale} Volta's plan of using the pile to
V charge batteries of Leyden jars.
The discovery of Nicholson and Carlisle made a great
impression on the mind of Humphry Davy (ft. 1778, d. 1829), a
young Cornishman who about this time was appointed Professor
of Chemistry at the E-oyal Institution in London. Davy at once
began to experiment vvitli Voltaic piles, and in November, 1800,§
showed that they give no current when the water between the
y pairs of plates is pure, and that their power of action is " in
great measure proportional to the power of the conducting
fluid substance between the double plates to oxydate the
zinc." This result, as he immediately perceived, did not
harmonize well with Volta's views on the source of electricity
in the pile, but was, on the other hand, in agreement with
, Eabroni's idea that galvanic effects are always accompanied by
chemical action. After a series of experiments he definitely
1 concluded that " the galvanic pile of Volta acts only when the
conducting substance between the plates is capable of oxydating
the zinc ; and that, in proportion as a greater quantity of
oxygen enters into combination with the zinc in a given time,
so in proportion is the power of the pile to decompose water
and to give the shock greater. It seems therefore reasonable
* Nicholson's Journal (4to), iv (1800), pp. 187,245: Phil. Mag., vii (1800),
p. 337.
t Phil. Mag., 1801, p. 427. J Phil. Mag., xii (1802), p. 161.
§ Nicholson's Journal (4to), iv (1800) ; Davy's Works, ii, p. 155.
Galvanism, from Galvani (o Ohm. 77
to conclude, though with our present quantity of facts we are
unable to explain the exact mode of operation, that the </
oxydatioii of the zinc in the pile, and the chemical changes
connected with it, are somehow the cause of the electrical effects ^
it produces." This principle of oxidation guided Davy in
designing many new types of pile, with elements chosen from
the whole range of the known metals.
Davy's chemical theory of the pile was supported by
Wollaston* and by Nicholson,f the latter of whom urged that
the existence of piles in which only one metal is used (with more
than one kind of fluid) is fatal to any theory which places the
seat of the activity in the contact of dissimilar metals.
Davy afterwards proposed J a theory of the voltaic pile
which combines ideas drawn from both the "contact" and
" chemical " explanations. Ho supposed that before the circuit
is closed, the copper and zinc disks in each contiguous pair
assume opposite electrostatic states, in consequence of inherent
"electrical energies" possessed by the metals; and when a >
communication is made between the extreme disks by a wire,
the opposite electricities annihilate each other, as in the dis-
charge of a Leyden jar. If the liquid (which Davy compared
to the glass of a Leyden jar) were incapable of decomposition,
the current would cease after this discharge. But the liquid in
the pile is composed of two elements which have inherent
attractions for electrified metallic surfaces : hence arises
chemical action, which removes from the disks the outermost
layers of molecules, whose energy is exhausted, and exposes
new metallic surfaces. The electrical energies of the copper and
zinc are consequently again exerted, and the process of electro-
motion continues. Thus the contact of metals is the cause
which disturbs the equilibrium, while the chemical changes
continually restore the conditions under which the contact
energy can be exerted.
In this and other memoirs Davy asserted that chemical
*Phil. Trans., 1801, p. 427. t Nicholson'* Journal, i (1802), p. 142.
; Phil. Trans., 1807, p. 1.
78 Galvanism, from Galvani to Ohm.
J affinity is essentially of an electrical nature. " Chemical and
electrical attractions," he declared,* "are produced by the
same cause, acting in one case on particles, in the other on
masses, of matter; and the same property, under different
modifications, is the cause of all the phenomena exhibited by
different voltaic combinations."
The further elucidation of this matter came chiefly from
- researches on electro-chemical decomposition, which we must
now consider.
A phenomenon which had greatly surprised Nicholson and
Carlisle in their early experiments was the appearance of
the products of galvanic decomposition at places remote from
each other. The first attempt to account for this was made in
1806 by Theodor von Grothussf (b. 1785, d. 1822) and by Davy,}
who advanced a theory that the terminals at which water is
decomposed have attractive and repellent powers ; that the pole
whence resinous electricity issues has the property of attracting
hydrogen and the metals, and of repelling oxygen and acid
substances, while the positive terminal has the power of attract-
ing oxygen and repelling hydrogen ; and that these forces are
sufficiently energetic to destroy or suspend the usual operation
of chemical affinity in the water-molecules nearest the
terminals. The force due to each terminal was supposed to
diminish with the distance from the terminal. When the
molecule nearest one of the terminals has been decomposed by
the attractive and repellent forces of the terminal, one of its
constituents is liberated there, while the other constituent, by
virtue of electrical forces (the oxygen and hydrogen being in
opposite electrical states), attacks the next molecule, which
is then decomposed. The surplus constituent from this attacks
the next molecule, and so on. Thus a chain of decompositions
and recompositions was supposed to be set up among the
molecules intervening between the terminals.
* Phil. Trans., 1826, p. 383. f Ann. de Cliim., Iviii (1806), p. 54.
t Bukerian lecture for 1806, Phil. Trans., 1807, p. 1. A theory similar to that
of Grothuss and Davy was communicated by Peter Mark Eoget (b. 1779, d. 1869)
in 1807 to the Philosophical Society of Manchester : cf. Roget's Galvanism, § 106.
Galvanism^ from ^Galvani to Ohm. 79
The hypothesis of Grothuss and Davy was attacked in 1825
by Aiiguste De La Kive* (6. 1801, d. 1873) of Geneva, on the
ground of its failure to explain what happens when different
liquids are placed in series in the circuit. If, for example, a
solution of zinc sulphate is placed in one compartment, and
water in another, and if the positive pole is placed in the
solution of zinc sulphate, and the negative pole in the water,
De La Rive found that oxide of zinc is developed round the
latter; although decomposition and recomposition of zinc
sulphate could not take place in the water, which contained
none of it. Accordingly, he supposed the constituents of the
decomposed liquid to be bodily transported across the liquids,
in close union with the moving electricity. In the electrolysis
of water, one current of electrified hydrogen was supposed to
leave the positive pole, and become decomposed into hydrogen
and electricity at the negative pole, the hydrogen being
there liberated as a gas. Another current in the same way
carried electrified oxygen from the negative to the positive
pole. In this scheme the chain of successive decompositions
imagined by Grothuss does not take place, the only molecules
decomposed being those adjacent to the poles.
The appearance of the products of decomposition at the
separate poles could be explained either in Grothuss' fashion
by assuming dissociations throughout the mass of liquid, or
in De La Rive's by supposing particular dissociated atoms
to travel considerable distances. Perhaps a preconceived
idea of economy in Nature deterred the workers of that time
from accepting the two assumptions together, when either of
them separately would meet the case. Yet it is to this apparent
redundancy that later researches have pointed as the truth.
Nature is what she is, and not what we would make her.
De La Rive was one of the most thoroughgoing opponents
of Volta's contact theory of the pile ; even in the case when
two metals are in contact in air only, without the intervention
* Annales de Cnimie, xxviii, 190.
80 Galvanism, from Galvani to Ohm.
of any liquid, he attributed the electric effect wholly to the
chemical affinity of the air for the metals.
During the long interval between the publication of the rival
hypotheses of Grothuss and De La Bive, little real progress
was made with the special problems of the cell ; but mean-
while electric theory was developing in other directions. One
of these, to which our attention will first be turned, was the
electro-chemical theory of the celebrated Swedish chemist,
Jons Jacob Berzelius (b. 1779, d. 1848).
Berzelius founded his theory,* which had been in one or two
of its features anticipated by Davy,f on inferences drawn from
Volta's contact effects. " Two bodies," he remarked, " which
have affinity for each other, and which have been brought into
mutual contact, are found upon separation to be in opposite
electrical states. That which has the greatest affinity for
oxygen usually becomes positively electrified, and the other
negatively."
This seemed to him to indicate that chemical affinity arises
from the play of electric forces, which in turn spring from
electric charges within the atoms of matter. To be precise,
he supposed each atom to possess two poles, which are the
seat of opposite electrifications, and whose electrostatic field is
the cause of chemical affinity.
By aid of this conception Berzelius drew a simple and vivid
picture of chemical combination. Two atoms, which are about
to unite, dispose themselves so that the positive pole of one
touches the negative pole of the other ; the electricities of these
two poles then discharge each other, giving rise to the heat and
light which are observed to accompany the act of combination.!
The disappearance of these leaves the compound molecule with
the two remaining poles ; and it cannot be dissociated into its
constituent atoms again until some means is found of restoring
to the vanished poles their charges. Such a means is afforded
* Memoirs of the Acad. of Stockholm, 1812 ; Nicholson's Journal of Nat. Phil.,
xxxiv (1813), 142, 153, 240, 319; xxxv, 38, 118, 159.
t Pnil. Trans., 1807. J This idea was Davy's.
Galvanism, from Gaivani to Ohm. 81
by the action of the galvanic pile in electrolysis : the opposite
electricities of the current invade the molecules of the
electrolyte, and restore the atoms to their original state of
polarization.
If, as Berzelius taught, all chemical compounds are formed
by the mutual neutralization of pairs of atoms, it is evident /
that they must have a binary character. Thus he conceived a
salt to be compounded of an acid and an oxide, and each of
these to be compounded of two other constituents. Moreover,
in any compound the electropositive member would be replace-
able only by another electropositive member, and the electro-
negative member only by another member also electronegative ;
so that the substitution of, e.g., chlorine for hydrogen in a
compound would be impossible — an inference which was
overthrown by subsequent discoveries in chemistry.
Berzelius succeeded in bringing the most curiously diverse
facts within the scope of his theory. Thus " the combination 1
of polarized atoms requires a motion to turn the opposite
poles to each other; and to this circumstance is owing the
facility with which combination takes place when one of the
two bodies is in the liquid state, or when both are in that
state ; and the extreme difficulty, or nearly impossibility, of
effecting an union between bodies, both of which are solid.
And again, since each polarized particle must have an electric
atmosphere, and as this atmosphere is the predisposing cause of
combination, as we have seen, it follows, that the particles
cannot act but at certain distances, proportioned to the
intensity of their polarity ; and hence it is that bodies, which
have affinity for each other, always combine nearly on the
instant when mixed in the liquid state, but less easily in the
gaseous state, and the union ceases to be possible under a
certain degree of dilatation of the gases ; as we know by the
experiments of Grothuss, that a mixture of oxygen and
hydrogen in due proportions, when rarefied to a certain
degree, cannot be set on fire at any temperature whatever." j
And again : " Many bodies require an elevation of temperature to
G
82 Galvanism, from Galvani to Ohm.
enable them to act upon each other. It appears, therefore,
that heat possesses the property of augmenting the polarity of
these bodies."
Berzelius accounted for Volta's electromotive series by
assuming the electrification at one pole of an atom to be some-
what more or somewhat less than what would be required to
neutralize the charge at the other pole. Thus each atom would
possess a certain net or residual charge, which might be of
either sign ; and the order of the elements in Volta's series
could be interpreted simply as the order in which they would
stand when ranged according to the magnitude of this residual
charge. As we shall see, this conception was afterwards
overthrown by Faraday.
Berzelius permitted himself to publish some speculations on
the nature of heat and electricity, which bring vividly before
us the outlook of an able thinker in the first quarter of the
nineteenth century. The great question, he says, is whether
v the electricities and caloric are matter or merely phenomena.
If the title of matter is to be granted only to such things as
are ponderable, then these problematic entities are certainly
not matter ; but thus to narrow the application of the term is,
he believes, a mistake; and he inclines to the opinion that
caloric is truly matter, possessing chemical affinities without
obeying the law of gravitation, and that light and all radiations
consist in modes of propagating such matter. This conclusion
makes it easier to decide regarding electricity. " From
the relation which exists between caloric and the electricities,"
he remarks, "it is clear that what may be true with regard
to the materiality of one of them must also be true with
regard to that of the other. There are, however, a quantity
of phenomena produced by electricity which do not admit of
explanation without admitting at the same time that electricity
is matter. Electricity, for instance, very often detaches
everything which covers the surface of those bodies which
conduct it. It, indeed, passes through conductors without
leaving any trace of its passage ; but it penetrates non-con-
Galvanism i from Galvani to Ohm. 83
ductors which oppose its course, and makes a perforation
precisely of the same description as would have been made
by something which had need of place for its passage. We
often observe this when electric jars are broken by an over-
charge, or when the electric shock is passed through a number
of cards, etc. We may therefore, at least with some proba-
bility, imagine caloric and the electricities to be matter,
destitute of gravitation, but possessing affinity to gravitating
bodies. When they are not confined by these affinities, they
tend to place themselves in equilibrium in the universe. The ^
suns destroy at every moment this equilibrium, and they send
the re-united electricities in the form of luminous rays towards
the planetary bodies, upon the surface of which the rays, being
arrested, manifest themselves as caloric ; and this last in its
turn, during the time required to replace it in equilibrium in
the universe, supports the chemical activity of organic and
inorganic nature."
It was scarcely to be expected that anything so speculative
as Berzelius' electric conception of chemical combination
would be confirmed in all particulars by subsequent discovery ;
and, as a matter of fact, it did not as a coherent theory survive
the lifetime of its author. But some of its ideas have
persisted, and among them the conviction which lies at its
foundation, that chemical affinities are, in the last resort, of *
electrical origin.
While the attention of chemists was for long directed to
the theory of Berzelius, the interest of electricians was
diverted from it by a discovery of the first magnitude in a
different region.
That a relation of some land subsists between electricity
and magnetism had been suspected by the philosophers of the
eighteenth century. The suspicion was based in part on some
curious effects produced by lightning, of a kind which may be
illustrated by a paper published in the Philosophical Transactions
in 1735.* A tradesman of Wakefield, we are told, "having put
*Phil. Trans, xxxix (1735), p. 74.
G 2
84 Galvanism, from Galvani to Ohm.
up a great number of knives and forks in a large box, and
having placed the box in the corner of a large room, there
happen'd in July, 1731, a sudden storm of thunder, lightning,
etc., by which the corner of the room was damaged, the Box
split, and a good many knives and forks melted, the sheaths
being untouched. The owner emptying the box upon a Counter
where some Nails lay, the Persons who took up the knives, that
lay upon the Nails, observed that the knives took up the Nails."
Lightning thus came to be credited with the power of
magnetizing steel ; and it was doubtless this which led Franklin*
in 1751 to attempt to magnetize a sewing-needle by means of
the discharge of Leyden jars. The attempt was indeed success-
ful ; but, as Van Marum afterwards showed, it was doubtful
whether the magnetism was due directly to the current.
More experiments followed. f In 1805 Jean Nicholas Pierre
Hachette (b. 1769, d. 1834) and Charles Bernard Desormes
(b. 1777, d. 1862) attempted to determine whether an insulated
voltaic pile, freely suspended, is oriented by terrestrial mag-
netism ; bat without positive result. In 1807 Hans Christian
Oersted (&. 1777, d. 1851), Professor of Natural Philosophy in
Copenhagen, announced his intention of examining the action
of electricity on the magnetic needle ; but it was not for some
years that his hopes were realized. If one of his pupils is to be
believed,* he was " a man of genius, but a very unhappy experi-
menter ; he could not manipulate instruments. He must
always have an assistant, or one of his auditors who had easy
hands, .to arrange the experiment."
During a course of lectures which he delivered in the winter
of 1819-20 on " Electricity, Galvanism, and Magnetism," the
idea occurred to him that the changes observed with the
compass-needle during a thunderstorm might give the clue to
the effect of which he was in search ; and this led him to think
that the experiment should be tried with the galvanic circuit
* Letter vi from Franklin to Collinson. f In 1774 the Electoral Academy
of Bavaria proposed the question, " Is there a real and physical analogy between
electric and magnetic forces ? " as the subject of a prize.
1 Cf. a letter from Hansteen inserted inBence Jones' Life of Faraday y ii, p. 395.
Galvanism, from Galvani to Ohm. 85
closed instead of open, and to inquire whether any effect is
produced on a magnetic needle when an electric current is
passed through a neighbouring wire. At first he placed the
wire at right angles to the needle, but observed no result.
After the end of a lecture in which this negative experiment
had been shown, the idea occurred to him to place the wire
parallel to the needle : on trying it, a pronounced deflexion was
observed, and the relation between magnetism and the electric
current was discovered. After confirmatory experiments with
more powerful apparatus, the public announcement was made
in July, 1820 *
Oersted did not determine the quantitative laws of the
;action, but contented himself with a statement of the qualita-
tive effect and some remarks on its cause, which recall the
magnetic speculations of Descartes : indeed, Oersted's concep-
tions may be regarded as linking those of the Cartesian school
to those which were introduced subsequently by Faraday. " To
the effect which takes place in the conductor and in the sur-
rounding space," he wrote, " we shall give the name of the
-conflict of electricity? " The electric conflict acts only on the
magnetic particles of matter. All non-magnetic bodies appear
penetrable by the electric conflict, while magnetic bodies, or
rather their magnetic particles, resist the passage of this conflict
Hence they can be moved by the impetus of the contending
powers.
" It is sufficiently evident from the preceding facts that the
•electric conflict is not confined to the conductor, but dispersed
pretty widely in the circumjacent space.
" From the preceding facts we may likewise collect, that this
conflict performs circles ; for without this condition, it seems
impossible that the one part of the uniting wire, when placed
below the magnetic pole, should drive it toward the east, and
when placed above it toward the west; for it is the nature of a
* Schweigger's Journal fur Chemie und Physik, zxix (1820), p. 275 ; Thomson's
Annals of Philosophy, xvi (1820), p. 273; Ostwald's Klattiter der
' Wi.ssenseha.ften, Nr. 63.
86 Galvanism, from Galvani to Ohm.
circle that the motions in opposite parts should have an opposite1
direction."
Oersted's discovery was described at the meeting of the
French Academy on September llth, 1820, by an academician
(Arago) who had just returned from abroad. Several investi-
gators in France repeated and extended his experiments ; and
the first precise analysis of the effect was published by two of
these, Jean-Baptiste Biot (b. 1774, d. 1862) and Felix Savart
(b. 1791, d. 1841), who, at a meeting of the Academy of Sciences
on October 30th, 1820, announced* that the action experienced
by a pole of austral or boreal magnetism, when placed at any
distance from a straight wire carrying a voltaic current, may be
4 thus expressed : " Draw from the pole a perpiendicular to the
wire ; the force on the pole is at right angles to this line and ta
the wire, and its intensity is proportional to the reciprocal of
the distance." This result was soon further analysed, the
attractive force being divided into constituents, each of which
was supposed to be due to some particular element of the
current ; in its new form the law may be stated thus : the-
magnetic force due to an element ds of a circuit, in which a
current i is flowing, at a point whose vector distance from ds is r,,
is (in suitable units)
i ids
— |ds,r|t or curl — .+
r3 J r
It was now recognized that a magnetic field may be produced
as readily by an electric current as by a magnet ; and, as Arago
soon showed,§ this, like any other magnetic field, is capable of
* Annales de Chimie, xv (1820), p. 222 ; Journal de Phys., xli, p. 51.
f If a and b denote two vectors, the vector whose components are (aybz — azby^
azb* — a*bz, axby — aybx) is called the vector product of a and b, and is denoted by
[a, b]. Its direction is at right angles to those of a and b, and its magnitude is
represented by twice the area of the triangle formed by them.
+ If a denotes any vector, the vector whose components are ^-z - -^, -—• - ^-*r
3% 9a* • -,
z-l - -— is denoted by curl a.
fo ty
§ Annales de Chimie, xv (1820), p. 93.
Galvanism, from Galvani to Ohm. 87
inducing magnetization in iron. The question naturally sug-
gested itself as to whether the similarity of properties between
currents and magnets extended still further, e.g. whether
conductors carrying currents would, like magnets, experience
ponderomotive forces when placed in a magnetic field, and
whether such conductors would consequently, like magnets,
exert ponderomotive forces on each other.
The first step towards answering these inquiries was taken
by Oersted* himself. " As," he said, " a body cannot put
another in motion without being moved in its turn, when it
possesses the requisite mobility, it is easy to foresee that the
galvanic arc must be moved by the magnet " ; and this he
verified experimentally.
The next step came from Andre Marie Ampere (b. 1775,
d. 1836), who at the meeting of the Academy on September 18th,
exactly a week after the news of Oersted's first discovery had
arrived, showed that two parallel wires carrying currents
attract each other if the currents are in the same direction,
and repel each other if the currents are in opposite directions.
During the next three years Ampere continued to prosecute
the researches thus inaugurated, and in 1825 published his
collected results in one of the most celebrated memoirsf in the
history of natural philosophy.
Ampere introduces his work by proclaiming himself a
follower of that school which explained all physical phenomena
in terms of equal and oppositely directed forces between pairs
of particles ; and he renounces the attempt to seek more
speculative, though possibly more fundamental, explanations
in terms of the motions of ultimate fluids and aethers. Never-
theless, he indicates two conceptions of this latter character, on
which such explanations might be founded.
In the firstj he suggests that the ponderomotive forces
* Schweigger's Journal fur Chem. u. Phys., xxix (1820), p. 364 ; Thomson's
Annals of Philosophy, xvi (1820), p. 375. t Mem. de 1'Acad., vi, p. 175.
% facueil tF observations electro- dynamiques, p. 215 ; and the memoir just cited,
pp. 285, 370.
88 Galvanism, from Galvani to Ohm.
between circuits carrying electric currents may be due to " the
reaction of the elastic fluid which extends throughout all
space, whose vibrations produce the phenomena of light," and
which is " put in motion by electric currents." This fluid or
aether can, he says, " be no other than that which results from
the combination of the two electricities/'
In the second conception,* Ampere suggests that the
interspaces between the metallic molecules of a wire which
carries a current may be occupied by a fluid composed of the
two electricities, not in the proportions which form the neutral
fluid, but with an excess of that one of them which is opposite
to the electricity peculiar to the molecules of the metal, and
which consequently masks this latter electricity. In this inter-
molecular fluid the opposite electricities are continually being
dissociated and recombined ; a dissociation of the fluid within
one inter-molecular interval having taken place, the positive
electricity thus produced unites with the negative electricity
of the interval next to it in the direction of the current, while
the negative electricity of the first interval unites with the
positive electricity of the next interval in the other direction.
Such interchanges, according to this hypothesis, constitute the
electric current.
Ampere's memoir is, however, but little occupied with the
more speculative side of the subject. His first aim was to
investigate thoroughly by experiment the ponderomotive forces
on electric currents.
" When," he remarks, " M. Oersted discovered the action
which a current exercises on a magnet, one might certainly have
suspected the existence of a mutual action between two circuits
carrying currents ; but this was not a necessary consequence ;
for a bar of soft iron also acts on a magnetized needle, although
there is no mutual action between two bars of soft iron."
Ampere, therefore, submitted the matter to the test of the
laboratory, and discovered that circuits carrying electric
currents exert ponderomotive forces on each other, and that
* Recucil d' observations electro-dunamiques, pp. 297, 300, 371.
Galvanism, from Gaivani to Ohm. 89
ponderomotive forces are exerted on such currents by magnets.
To the science which deals with the mutual action of currents
he gave the name electro-dynamics ;* and he showed that the
action obeys the following laws : —
(1) The effect of a current is reversed when the direction of
the current is reversed.
(2) The effect of a current flowing in a circuit twisted into
small sinuosities is the same as if the circuit were smoothed out.
(3) The force exerted by a closed, circuit on an element of
another circuit is at right angles to the latter.
(4) The force between two elements of circuits is unaffected
when all linear dimensions are increased proportionately, the
current-strengths remaining unaltered.
From these data, together with his assumption that the force
between two elements of circuits acts along the line joining them,
Ampere obtained an expression of this force : the deduction may
be made in the following way : —
Let ds, ds' be the elements, r the line joining them, and i, i'
the current-strengths. From (2) we see that the effect of ds on
ds' is the vector sum of the effects of dx, dy, dz on ds', where
these are the three components of ds: so the required force
must be of the form —
r x a scalar quantity which is linear and homogeneous in ds ;
and it must similarly be linear and homogeneous in ds' ; so
using (1), we see that the force must be of the form
F = ill | (ds . ds') 4> (r) + (ds . r) (ds'. r) i/, (r)} ,
where <£ and i// denote undetermined functions of r.
From (4) it follows that when ds, ds', r are all multiplied by
the same number, F is unaffected : this shows that
4>(r) = - and f (r) = - ,
where A and B denote constants. Thus we have
, M(ds.ds') £(ds.r)(ds'. r))
F = n r \ + - — —- — ; •
( r3 r6 )
*. Loc. cit., p. 298.
90 Galvanism , from Galvani to Ohm.
Now, by (3), the resolved part of F along ds' must vanish when
integrated round the circuit s, i.e. it must be a complete
differential when dr is taken to be equal to - ds. That is to-
say,
^(ds.ds')(r.ds') £(ds . r) (ds'. r)2
/o-»3 .f\>£
must be a complete differential ; or
must be a complete differential ; and therefore
7 A BiA
d'^ = --5(dS'r)>
3^ B J
or ~2^" dr = r*dT'
or B = - I A.
Thus finally we have
F = Constant x ii'i || (ds . ds') - -5 (ds . r)(ds'. r)
This is Ampere's formula : the multiplicative constant depends
of course on the units chosen, and may be taken to be - 1.
The weakness of Ampere's work evidently lies in the
assumption that the force is directed along the line joining the-
two elements : for in the analogous case of the action between
two magnetic molecules, we know that the force is not directed
along the line joining the molecules. It is therefore of interest
to find the form of F when this restriction is removed.
For this purpose we observe that we can add to the expression
already found for F any term of the form
0(r) . (ds . r) . ds',
where 0(r) denotes any arbitrary function of r ; for since
this term vanishes when integrated round the circuit s ; and it
Galvanism, from Galvani to Ohm. 91
contains ds and ds' linearly and homogeneously, as it should.
We can also add any terms of the form
rf{r..(ds'.r).x(r)|,
where \(r] denotes any arbitrary function of r, and d denotes
differentiation along the arc s, keeping ds' fixed (so that
dr = - ds) ; this differential may be written
- ds . (ds'. r) . x(r) - rx(r) (ds'. ds) - * x'(r) r (ds . r) (ds'. r).
In order that the law of Action and Eeaction may not be
violated, we must combine this with the former additional term
so as to obtain an expression symmetrical in ds and ds' : and
hence we see finally that the general value of F is given by the
equation
F = -n'rjj|(ds.ds')-J(ds.r)(ds.r)j
+ x(»-; (ds' . r) ds + x(r) (ds . r) . ds' + x(r) (ds . ds')r
+ ix'(r)(ds.r)(ds'.r)r.
The simplest form of this expression is obtained by taking
when we obtain
• •/
F = - {(ds . r) . ds' + (ds'. r)ds - (ds . ds')r} .
The comparatively simple expression in brackets is the
vector part of the quaternion product of the three vectors
ds, r, ds'.*
From any of these values of F we can find the ponderomotive
force exerted by the whole circuit s on the element ds' : it is, in
fact, from the last expression,
u'f1
[?-3((ds'.r).ds-(ds.ds>},
* The simpler form of F given in the text is, if the term in da' be omitted, the
form given by Grassmann, Ann. d. Phys. Ixiv (1845), p. 1. For further work on
this subject cf. Tait, Proc. R. S. Edin. viii (1873), p. 220, and Korteweg, Journal
fiir Math, xc (1881), p. 45.
92 Galvanism, from Galvani to Ohm.
or i [ds'. B],
where B =
Now this value of B is precisely the value found by Biot and
Savart* for the" magnetic intensity at ds' due to the- current i in
the circuit s. Thus we see that the ponderomotive force on a
current-element ds' in a magnetic field B is i' [ds'. B].
Ampere developed to a considerable extent the theory
of the equivalence of magnets with circuits carrying currents;
and showed that an electric current is equivalent, in its
magnetic effects, to a distribution of magnetism on any
surface terminated by the circuit, the axes of the magnetic
molecules being everywhere normal to this surface :f such a
magnetized surface is called a mayiwtic shell. He preferred,
I however, to regard the current rather than the magnetic fluid
as the fundamental entity, and considered magnetism to be
really an electrical phenomenon : each magnetic molecule owes
its properties, according to this view, to the presence within it
/ of a small closed circuit in which an electric current is
perpetually flowing.
The impression produced by Ampere's memoir was great
and lasting. Writing half a century afterwards, Maxwell
speaks of it as " one of the most brilliant achievements in
science." " The whole," he says, " theory and experiment,
seems as if it had leaped, full-grown and full-armed, from the
brain of the ' Newton of electricity/ It is perfect in form and
unassailable in accuracy ; and it is summed up in a formula
from which all the phenomena may be deduced, and which
must always remain the cardinal formula of electrodynamics."
Not long after the discovery by Oersted of the connexion
between galvanism and magnetism, a connexion was discovered
between galvanism and heat.| In 1822 Thomas Johann Seebeck
* See ante, p. 86. t Loc. cit., p. 367.
Galvanism, from ^Galvani to Ohm. 93
(b. 1770, d. 1831), of Berlin discovered* that an electric current
can be set up in a circuit of metals, without the interposition
of any liquid, merely by disturbing the equilibrium of
temperature. Let a ring be formed of copper and bismuth
soldered together at the two extremities; to establish a
current it is only necessary to heat the ring at one of these
junctions. To this new class of circuits the name thermo-
electric was given.
It was found that the metals can be arranged as a
thermo-electric series, in the order of their power of generating
currents when thus paired, and that this order is quite different
from Volta's order of electromotive potency. Indeed antimony
and bismuth, which are near each other in the latter series, are
at opposite extremities of the former.
The currents generated by thermo-electric means are
generally feeble : and the mention of this fact brings us to
the question, which was about this time engaging attention,
of the efficacy of different voltaic arrangements.
Comparisons of a rough kind had been instituted soon after
the discovery of the pile. The French chemists Antoine
FranQois de Fourcroy (b. 1755, d. 1809), Louis Mcolas
Yauquelin (b. 1763, d. 1829), and Louis Jacques Thenard
(b. 1777, d. 1857) foundf in 1801, on varying the size of the
metallic disks constituting the pile, that the sensations
produced on the human frame were unaffected so long as the
number of disks remained the same; but that the power 'of
burning finely drawn wire was altered; and that the latter
power was proportional to the total surface of the disks
employed, whether this were distributed among a small number
of large disks, or a large number of small ones. This was
* Abhandl. d. Berlin Akad. 1822-3 ; Ann. d. Phys. Ixxiii (1823), pp. 115,
430 ; vi (1826), pp. 1, 133, 253.
Volta had previously noticed that a silver plate whose ends were at different
temperatures appeared to act like a voltaic cell.
Further experiments were performed by James Gumming (£. 1777, d. 1861),
Professor of Chemistry at Cambridge, Trans. Camb. Phil. Soc. ii (1823), p. 47,
and by Antoine Cesar Becquerel (b. 1788, d. 1878), Annales de Chimie, xxxi
(1826), p. 371. t Ann. de Chimie, xxxix (1801), p. 103.
94 Galvanism, from Galvani to Ohm.
explained by supposing that small plates give a small quantity
of the electric fluid with a high velocity, while large plates
give a larger quantity with no greater velocity. Shocks,
which were supposed to depend on the velocity of the fluid
alone, would therefore not be intensified by increasing the size
of the plates.
The effect of varying the conductors which connect the
terminals of the pile was also studied. Nicolas Grautherot
(b. 1753, d. 1803) observed* that water contained in tubes which
have a narrow opening does not conduct voltaic currents so
well as when the opening is more considerable. This experi-
ment is evidently very similar to that which Beccaria had
performed half a century previously! with electrostatic
discharges.
As we have already seen, Cavendish investigated very
-completely the power of metals to conduct electrostatic
discharges; their power of conducting voltaic currents was
now examined by Davy.J His method was to connect the
terminals of a voltaic battery by a path containing water
(which it decomposed), and also by an alternative path
consisting of the metallic wire under examination. When the
length of the wire was less than a certain quantity, the water
ceased to be decomposed ; Davy measured the lengths and
weights of wires of different materials and cross-sections under
these limiting circumstances ; and, by comparing them, showed
that the conducting power of a wire formed of any one metal
is inversely proportional to its length and directly proportional
to its sectional area, but independent of the shape of the cross-
section.! The latter fact, as he remarked, showed that voltaic
currents pass through the substance of the conductor and not
along its surface.
Davy, in the same memoir, compared the conductivities of
various metals, and studied the effect of temperature : he found
* Annales de China., xxxix (1801), p. 203. t See p. 53.
% Phil. Trans., 1821, p. 433. His results were confirmed afterwards by
Becquerel, Annales de Chiiuie, xxxii (1825), p. 423.
6 These results had been known to Cavendish.
Galvanism , from Galvani to Ohm. 95
that the conductivity varied with the temperature, being
" lower in some inverse ratio as the temperature was higher."
He also observed that the same magnetic power is exhibited
by every part of the same circuit, even though it be formed
of wires of different conducting powers pieced into a chain,
so that " the magnetism seems directly as the quantity of
electricity which they transmit."
The current which flows in a given voltaic circuit evidently
depends not only on the conductors which form the circuit,
but also on the driving-power of the battery. In order to form
a complete theory of voltaic circuits, it was therefore necessary
to extend Davy's laws by taking the driving-power into
account. This advance was effected in 1826 by Georg Simom
Ohm* (b. 1787, d. 1854).
Ohm had already carried out a considerable amount of
experimental work on the subject, and had, e.g., discovered that
if a number of voltaic cells are placed in series in a circuit, the
current is proportional to their number if the external
resistance is very large, but is independent of their number if
the external resistance is small. He now essayed the task
of combining all the known results into a consistent theory.
For this purpose he adopted the idea of comparing the flow
of electricity in a current to the flow of heat along a wire, the
theory of which had been familiar to all physicists since the
publication of Fourier's Theorie analytique de la chcdeur in
1822. " I have proceeded," he says, " from the supposition that
the communication of the electricity from one particle takes
place directly only to the one next to it, so that no immediate
transition from that particle to any other situate at a greater
distance occurs. The magnitude of the flow between two
adjacent particles, under otherwise exactly similar circum-
stances, I have assumed to be proportional to the difference of
*Ann. d. Phys. vi (1826), p. 459 ; vii, pp. 45,117; Die Galvanische Eette
mathematisch bearbeitet : Berlin, 1827 ; translated in Taylor's Scientific Memoirs,
ii (1841), p. 401. Cf. also subsequent papers by Ohm in Kastner's Archiv fur
d. ges. Naturkhre, and Schweigger's Jahrbuch.
96 • Galvanism, from Galvani to Ohm.
the electric forces existing in the two particles ; just as, in the
theory of heat, the flow of caloric between two particles is
regarded as proportional to the difference of their temperatures."'
'*' The comparison between the flow of electricity and the flow
of heat suggested the propriety of introducing a quantity
whose behaviour in electrical problems should resemble that of
temperature in the theory of heat. The differences in the
values of such a quantity at two points of a circuit would
provide what was so much needed, namely, a measure of the
"driving-power" acting on the electricity between these
points. To carry out this idea, Ohm recurred to Volta's theory
of the electrostatic condition of the open pile. It was cus-
tomary to measure the " tension " of a pile by connecting one
terminal to earth and testing the other terminal by an
electroscope. Accordingly Ohm says : " In order to investigate
the changes which occur in the electric condition of a body A
in a perfectly definite manner, the body is each time brought,
under similar circumstances, into relation with a second
moveable body of invariable electrical condition, called the
electroscope ; and the force with which the electroscope is
repelled or attracted by the body is determined. This force is
termed the electroscopic force of the body A"
" The same body A may also serve to determine the electro-
scopic force in various parts of the same body. For this
purpose take the body A of very small dimensions, so that
when we bring it into contact with the part to be tested of any
third body, it may from its smallness be regarded as a substitute
for this part : then its electroscopic force, measured in the way
described, will, when it happens to be different at the various
places, make known the relative differences with regard to
electricity between these places."
Ohm assumed, as was customary at that period, that when
two metals are placed in contact, " they constantly maintain at
the point of contact the same difference between their electro-
scopic forces." He accordingly supposed that each voltaic cell
possesses a definite tension, or discontinuity of electroscopic
Galvanism, from Galvani to Ohm. 97
force, which is to be regarded as its contribution to the driving-
force of any circuit in which it may be placed. This assumption
confers a definite meaning on his use of the term " electroscopic
force " ; the force in question is identical with the electrostatic
potential. But Ohm and his contemporaries did not correctly
understand the relation of galvanic conceptions to the j
electrostatic functions of Poisson. The electroscopic force
in the open pile was generally identified with the thickness
of the electrical stratum at the place tested ; while Ohm,
recognizing that electric currents are not confined to the
surface of the conductors, but penetrate their substance,
seems to have thought of the electroscopic force at a place in
a circuit as being proportional to the volume-density of
electricity there — an idea in which he was confirmed by the
relation which, in an analogous case, exists between the
temperature of a body and the volume-density of heat
supposed to be contained in it.
Denoting, then, by S the current which flows in a wire of
conductivity y, when the difference of the electroscopic forces at
the terminals is E, Ohm writes
S = yE.
From this formula it is easy to deduce the laws already given
by Davy. Thus, if the area of the cross-section of a wire
is Ay we can by placing n such wires side by side construct
a wire of cross-section nA. If the quantity E is the same
for each, equal currents will flow in the wires ; and therefore
the current in the compound wire will be ?i times that in
the single wire ; so when the quantity E is unchanged, the
current is proportional to the cross-section; that is, the
conductivity of a wire is directly proportional to its cross-section,
which is one of Davy's laws.
In spite of the confusion which was attached to the idea of
electroscopic force, and which was not dispelled for some years,
the publication of Ohm's memoir marked a great advance
in electrical philosophy. It was now clearly understood that
the current flowing in any conductor depends only on the
H
98 Galvanism^ from Galvani to Ohm.
conductivity inherent in the conductor and on another variable
which bears to electricity the same relation that temperature
bears to heat ; and, moreover, it was realized that this latter
variable is the link connecting the theory of currents with
the older theory of electrostatics. These principles were a
sufficient foundation for future progress; and much of the
work which was published in the second quarter of the century
was no more than the natural development of- the principles
laid down by Ohm.*
It is painful to relate that the discoverer had long to wait
before the merits of his great achievement were officially
recognized. Twenty- two years after the publication of the
memoir on the galvanic circuit, he was promoted to a university
professorship ; this he held for the five years which remained
until his death in 1854.
* Ohm's theory was confirmed experimentally by several investigators, among
whom may be mentioned Gustav Theodor Feehner(i. 1801, d. 1887) (Maassbestim-
mungen iiber die Galvanische Kette, Leipzig, 1831), and Charles Wheatstone
(b. 1802, d. 1875) (Phil. Trans,, 1843, p. 303).
CHAPTER IV.
THE LUMINIFEROUS MEDIUM, FROM BRADLEY TO FRESNEL.
ALTHOUGH Newton, as we have seen, refrained from committing
himself to any doctrine regarding the ultimate nature of light,
the writers of the next generation interpreted his criticism of
the wave-theory as equivalent to an acceptance of the
corpuscular hypothesis. As it happened, the chief optical
discovery of this period tended to support the latter theory,
by which it was first and most readily explained. In 1728
James Bradley (b. 1692, d. 1762), at that time Savilian
Professor of Astronomy at Oxford, sent to the Astronomer
Royal (Halley) an " Account of a new discovered motion of the
Fix'd Stars."* In observing the star y in the head of the
Dragon, he had found that during the winter of 1725-6 the
transit across the meridian was continually more southerly,
while during the following summer its original position was
restored by a motion northwards. Such an effect could not be
explained as a result of parallax ; and eventually Bradley
guessed it to be due to the gradual propagation of light.f
Thus, let CA denote a ray of light, falling on the line BA ;
and suppose that the eye of the observer is travelling ^
along BA, with a velocity which is to the velocity
of light as BA is to CA. Then the corpuscle of
light, by which the object is discernible to the eye
at A, would have been at C when the eye was at
B. The tube of a telescope must therefore be pointed
in the direction BC, in order to receive the rays
from an object whose light is really propagated in
the direction CA. The angle BCA measures the
difference between the real and apparent positions »
of the object ; and it is evident from the figure that the sine of
•Phil. Trans, xxxv (1728), p. 637.
t Roemer, in a letter to Huygens of date 30th Dec., 1677, mentions a suspected
displacement of the apparent position of a star, due to the motion of the earth at
right angles to the line of sight. Cf . Correspondance de Huygens, viii, p. 53.
H 2
100 The Lumini/erous Medium,
this angle is to the sine of the visible inclination of the object
to the line in which the eye is moving, as the velocity of the eye
is to the velocity of light. Observations such as Bradley's will
therefore enable us to deduce the ratio of the mean orbital
velocity of the earth to the velocity of light, or, as it is called,.
the constant of 'aberration ; from its value Bradley calculated that
light is propagated from the sun to the earth in 8 minutes
12 seconds, which, as he remarked, "is as it were a Mean
betwixt what had at different times been determined from the
eclipses of Jupiter's satellites."*
With the exception of Bradley's discovery, which was
primarily astronomical rather than optical, the eighteenth
century was decidedly barren, as regards both the experimental
and the theoretical investigation of light ; in curious contrast
to the brilliance of its record in respect of electrical researches.
But some attention must be given to a suggestive study f of the
aether, for which the younger John Bernoulli (b. 1710, d. 1790)
was in 1736 awarded the prize of the French Academy. His
ideas seem to have been originally suggested by an attempt};
*Struve in 1845 found for the constant of aberration the value 20"'445, which
lie afterwards corrected to 20"'463. This was superseded in 1883 by the value
20"-492, determined by M. Nyren. The observations of both Struve and Nyren
were made with the transit in the prime vertical. The method now generally
used depends on the measurement of differences of meridian zenith distances
(Talcott's method, as applied by F. Kiistner, Beobachtungs-Ergebnisse der kon.
Stern warte zu Berlin, Heft 3, 1888) ; the value at present favoured for the
constant of aberration is 20"-523. Cf. Chandler, Ast. Journal, xxiii, pp. 1, 12
(1903).
The collective translatory motion of the solar system gives rise to aberrational:
terms in the apparent places of the fixed stars ; but the principal term of this
character does not vary with the time, and consequently is equivalent to a
permanent constant displacement. The second-order terms (i.e. those which
involve the ordinary constant of aberration multiplied by the sun's velocity)
might be measurable quantities in the case of stars near the Pole ; and the same is
true of the variations in the first-order terms (i.e. those which involve the sun's
velocity not multiplied by the constant of aberration) due to the circumstance
that the star's apparent R. A. and Declination, which occur in these terms, are
not constant, but are affected by Precession, Nutation, and Aberration. Cf.
Seeliger, Ast. Nach., cix., p. 273 (1884).
t Printed in 1752, in the Recueil des pieces qui ont remportes les prix de V Acad.y.
tome iii. J Acta eniditorum, MDCCI, p. 19.
from Bradky to Fresnel. 101
which his father, the elder John Bernoulli (b. 1667, d. 1748),
had made in 1701 to connect the law of refraction with the
mechanical principle of the composition of forces. If two
opposed forces whose ratio is ju maintain in equilibrium a
particle which is free to move only in a given plane, it follows
from the triangle of forces that the directions of the forces must
obey the relation
sin i = fj. sin r,
where i and r denote the angles made by these directions with
the normals to the plane. This is the same equation as that
which expresses the law of refraction, and the elder Bernoulli
conjectured that a theory of light might be based on it ; but
he gave no satisfactory physical reason for the existence of
forces along the incident and refracted rays. This defect his
son now proceeded to remove.
All space, according to the younger Bernoulli, is permeated
by a fluid aether, containing an immense number of excessively
small whirlpools. The elasticity which the aether appears to
possess, and in virtue of which it is able to transmit vibrations,
is really due to the presence of these whirlpools ; for, owing to
-centrifugal force, each whirlpool is continually striving to
dilate, and so presses against the neighbouring whirlpools. It
will be seen that Bernoulli is a thorough Cartesian in spirit ;
not only does he reject action at a distance, but he insists that
•even the elasticity of his aether shall be explicable in terms of
matter and motion.
This aggregate of small vortices, or " fine-grained turbulent
motion," as it came to be called a century and a half later,* is
interspersed with solid corpuscles, whose dimensions are small
-compared with their distances apart. These are pushed about
by the whirlpools whenever the aether is disturbed, but never
travel far from their original positions.
A source of light communicates to its surroundings a
disturbance which condenses the nearest whirlpools ; these by
* Cf . Lord Kelvin's vortex-sponge aether, described later in this work.
102 The Luminiferous Medium i
their condensation displace the contiguous corpuscles from their
equilibrium position ; and these in turn produce condensations
in the whirlpools next beyond them, so that vibrations are
propagated in every direction from the luminous point. It is
curious that Bernoulli speaks of these vibrations as longitudinal,
and actually contrasts them with those of a stretched cord,
which, " when it is slightly displaced from its rectilinear form,
and then let go, performs transverse vibrations in a direction at
right angles to the direction of the cord." When it is
remembered that the objection to longitudinal vibrations, on
the score of polarization, had already been clearly stated by
Newton, and that Bernoulli's aether closely resembles that
which Maxwell invented in 1861-2 for the express purpose of
securing transversality of vibration, one feels that perhaps no
man ever so narrowly missed a great discovery.
Bernoulli explained refraction by combining these ideas
with those of his father. Within the pores of ponderable
bodies the whirlpools are compressed, so the centrifugal force
must vary in intensity from one medium to another. Thus a
corpuscle situated in the interface between two media is acted
on by a greater elastic force from one medium than from the
other; and by applying the triangle of forces to find the-
conditions of its equilibrium, the law of Snell and Descartes
r may be obtained.
Not long after this, the echoes of the old controversy
' between Descartes and Fermat about the law of refraction
were awakened* by Pierre Louis Moreau deMaupertuis (b. 1698,,
d. 1759).
It will be remembered that according to Descartes the
velocity of light is greatest in dense media, while according to-
Fermat the propagation is swiftest in free aether. The argu-
ments of the corpuscular theory convinced Maupertuis that on
this particular point Descartes was in the right ; but never-
theless he wished to retain for science the beautiful method by
which Fermat had derived his result. This he now proposed
*Mem. de 1'Acad., 1744, p. 417.
from Bradley to Fresnel. 103
to do by modifying Fermat's principle so as to make it agree
with the corpuscular theory; instead of assuming that light
follows the quickest path, he supposed that " the path described
is that by which the quantity of action is the least " ; and this
action he defined to be proportional to the sum of the spaces
described, each multiplied by the velocity with which it is
traversed. Thus instead of Fermat's expression
dt or
tds
} v
(where t denotes time, v velocity, and ds an element of the path)
Maupertuis introduced
/v ds
as the quantity which is to assume its minimum value when the
path of integration is the actual path of the light. Since
Maupertuis' v, which denotes the velocity according to the
corpuscular theory, is proportional to the reciprocal of Fermat's
v, which denotes the velocity according to the wave- theory, the
two expressions are really equivalent, and lead to the same law
of refraction. Maupertuis' memoir is, however, of great
interest from the point of view of dynamics ; for his suggestion
was subsequently developed by himself and by Euler and
Lagrange into a general principle which covers the whole
range of Nature, so far as Nature is a dynamical system.
The natural philosophers of the eighteenth century for the
most part, like Maupertuis, accepted the corpuscular hypothesis ;
'but the wave-theory was not without defenders. Franklin*
declared for it ; and the celebrated mathematician Leonhard
Euler (b. 1707, d. 1783) ranged himself on the same side. In a
work entitled Nova Theoria Lucis et Colorum, published! while
he was living under the patronage of Frederic the Great at
Berlin, he insisted strongly on the resemblance between light
and sound ; " light is in the aether the same thing as sound in
air/' Accepting Newton's doctrine that colour depends on
* Letter xxiii, written in 1752.
tL. Euleri Opuscula varii argumenti, Berlin, 1746, p. 169.
104 The Luminiferous Medium ,
wave-length, he in this memoir supposed the frequency greatest
for red light, and least for violet ; but a few years later* he
adopted the opposite opinion.
The chief novelty of Euler's writings on light is his
explanation of the manner in which material bodies appear
coloured when viewed by white light ; and, in particular, of the
way in which the colours of thin plates are produced. He
denied that such colours are due to a more copious reflexion of
light of certain particular periods, and supposed that they
represent vibrations generated within the body itself under the
stimulus of the incident light. A coloured surface, according
to this hypothesis, contains large numbers of elastic molecules,
which, when agitated, emit light of period depending only
on their own structure. The colours of thin plates Euler
explained in the same way ; the elastic response and free period
of the plate at any place would, he conceived, depend on its
thickness at that place ; and in this way the dependence of the
colour on the thickness was accounted for, the phenomena as
a whole being analogous to well-known effects observed in
experiments on sound.
An attempt to improve the corpuscular theory in another
direction was made in 1752 by the Marquis de Courtivron,f and
independently in the following year by T. Melville These
writers suggested, as an explanation of the different refran-
gibility of different colours, that " the differently colour'd rays
are projected with different velocities from the luminous body :
the red with the greatest, violet with the least, and the inter-
mediate colours with intermediate degrees of velocity." On
this supposition, as its authors pointed out, the amount of
aberration would be different for every different colour ; and
the satellites of Jupiter would change colour, from white through
green to violet, through an interval of more than half a minute
before their immersion into the planet's shadow ; while at
emersion the contrary succession of colours should be observed,
* Mem. del' Acad.de Berlin, 1752, p. 262. t Courtivron's Traite cfoptique, 1752.
JPhil. Trans, xlviii (1753), p. 262.
from Bradley to FremeL 105
beginning with red and ending in white. The testimony of
practical astronomers was soon given that such appearances are
not observed ; and the hypothesis was accordingly abandoned.
The fortunes of the wave-theory began to brighten at the
end of the century, when a new champion arose. Thomas
Young, born at Milverton in Somersetshire in 1773, and
trained to the practice of medicine, began to write on optical
theory in 1799. In his first paper* he remarked that, according'1
to the corpuscular theory, the velocity of emission of a
corpuscle must be the same in all cases, whether the projecting
force be that of the feeble spark produced by the friction of two Q
pebbles, or the intense heat of the sun itself — a thing almost
incredible. This difficulty does not exist in the undulatory
theory, since all disturbances are known to be transmitted^
through an elastic fluid with the same velocity. The reluctance
which some philosophers felt to filling all space with an elastic
fluid he met with an argument which strangely foreshadows
the electric theory of light : " That a medium resembling in
many properties that which has been denominated ether does
really exist, is undeniably proved by the phenomena of
electricity. The rapid transmission of the electrical shock
shows that the electric medium is possessed of an elasticity as
great as is necessary to be supposed for the propagation of light.
Whether the electric ether is to be considered the same with
the luminous ether, if such a fluid exists, may perhaps at some
future time be discovered by experiment : hitherto I have not
been able to observe that the refractive power of a fluid
undergoes any change by electricity."
Young then proceeds to show the superior power of the^
wave-theory to explain reflexion and refraction. In the
corpuscular theory it is difficult to see why part of the light
should be reflected and another part of the same beam reflected ;
but in the undulatory theory there is no trouble, as is shown
by analogy with the partial reflexion of sound from a cloud or _,
denser stratum of air : " Nothing more is necessary than to
* Phil. Tni™., 1800, p. 106.
106 The JLuminiferous Medium,
suppose all refracting media to retain, by their attraction, a,
greater or less quantity of the luminous ether, so as to make its-
density greater than that which it possesses in a vacuum,
without increasing its elasticity." This is precisely the
hypothesis adopted later by Fresnel and Green.
In 1801 Young made a discovery of the first magnitude*
when attempting to explain Newton's rings on the principles of
the wave-theory. Eejecting Euler's hypothesis of induced
vibrations, he assumed that the colours observed all exist in
the incident light, and showed that they could be derived from
it by a process which was now for the first time recognized
in optical science.
The idea of this process was not altogether new, for it had
been used by Newton in his theory of the tides. " It may
happen," he wrote, f " that the tide may be propagated from the
ocean through different channels towards the same port, and
may pass in less time through some channels than through
others, in which case the same generating tide, being thus
divided into two or more succeeding one another, may produce
by composition new types of tide." Newton applied this-
principle to explain the anomalous tides at Batsha in Tonkin,
which had previously been described by Halley.J
Young's own illustration of the principle is evidently
suggested by Newton's. " Suppose," he says,§ " a number of
equal waves of water to move upon the surface of a stagnant
lake, with a certain constant velocity, and to enter a narrow
channel leading out of the lake ; suppose then another similar
cause to have excited another equal series of waves, which
arrive at the same channel, with the same velocity, and at the
same time with the first. Neither series of waves will destroy
the other, but their effects will be combined ; if they enter the
channel in such a manner that the elevations of one series
coincide with those of the other, they must together produce a
series of greater joint elevations ; but if the elevations of one
* Phil. Trans., 1802, pp. 12, 387. t Principia, Book in, Prop. 24.
% Phil. Trans, xiv (1684), p. 681. § Young's Works, i, p. 202.
from Bradley to Fremel. 107
series are so situated as to correspond to the depressions of the
other, they must exactly fill up those depressions, and the
surface of the water must remain smooth. Now I maintain
that similar effects take place whenever two portions of light
are thus mixed ; and this I call the general law of the interference
of light."
Thus, " whenever two portions of the same light arrive to the
eye by different routes, either exactly or very nearly in the same
direction, the light becomes most intense when the difference of
the routes is any multiple of a certain length, and least intense
in the intermediate state of the interfering portions ; and this
length is different for light of different colours."
Young's explanation of the colours of thin plates as seen by
reflexion was, then, that the incident light gives rise to two
beams which reach the eye : one of these beams has been
reflected at the first surface of the plate, and the other at the
second surface ; and these two beams produce the colours by
their interference.
One difficulty encountered in reconciling this theory with
observation arose from the fact that the central spot in Newton's
rings (where the thickness of the thin Him of air is zero) is
black and not white, as it would be if the interfering beams were
similar to each other in all respects. To account for this Young '
showed, by analogy with the impact of elastic bodies, that when ->
light is reflected at the surface of a denser medium, its phase
is retarded by half an undulation : so that the interfering
beams at the centre of Newton's rings destroy each other. The
correctness of this assumption he verified by substituting essence
of sassafras (whose refractive index is intermediate between those
of crown and flint glass) for air in the space between the lenses ;
as he anticipated, the centre of the ring-system was now white.
Newton had long before observed that the rings are smaller ~*
when the medium producing them is optically more dense.
Interpreted by Young's theory, this definitely proved that the
wave-length of light is shorter in dense media, and therefore ^
that its velocity is less.
108 The Lumini/erous Medium,
The publication of Young's papers occasioned a fierce attack
on him in the Edinburgh Review, from the pen of Henry
Brougham, afterwards Lord Chancellor of England. Young
replied in a pamphlet, of which it is said* that only a single
copy was sold ; and there can be no doubt that Brougham for
the time being achieved his object of discrediting the wave-
theory, f
Young now turned his attention to the fringes of shadows.
In the corpuscular explanation of these, it was supposed that
the attractive forces which operate in refraction extend their
influence to some distance from the surfaces of bodies, and
inflect such rays as pass close by. If this were the case, the
amount of inflexion should obviously depend on the strength of
the attractive forces, and consequently on the refractive indices
of the bodies — a proposition which had been refuted by the
experiments of s'Gravesande. The cause of diffraction effects
was thus wholly unknown, until Young, in the Bakerian lecture
for 1803,J showed that the principle of interference is concerned
in their formation ; for when a hair is placed in the cone of rays
diverging from a luminous point, the internal fringes (i.e. those
within the geometrical shadow) disappear when the light passing
on one side of the hair is intercepted. His conjecture as to the
origin of the interfering rays was not so fortunate ; for he attri-
buted the fringes outside the geometrical shadow to interference
between the direct rays and rays reflected at the diffracting
edge ; and supposed the internal fringes of the shadow of a
narrow object to be due to the interference of rays inflected by
the two edges of the object.
The success of so many developments of the wave-theory
led Young to inquire more closely into its capacity for solving
the chief outstanding problem of optics — that of the behaviour
of light in crystals. The beautiful construction for the extra-
* Peacock's Life of Young.
t" Strange fellow," wrote Macaulay, when half a century afterwards he
found himself sitting beside Brougham in the House of Lords, " his powers
gone : his spite immortal."
I Phil. Trans., 1804; Young's Works, i, p. 179.
from Bradley ( to FresneL 109
ordinary ray given by Huygens had lain neglected for a century ;
and the degree of accuracy with which it represented the
observations was unknown. At Young's suggestion Wollaston*
investigated the matter experimentally, and showed that the
agreement between his own measurements and Huygens' rule
was remarkably close. " I think," he wrote, " the result must be
admitted to be highly favourable to the Huygeniaii theory ;
and, although the existence of two refractions at the same time,
in the same substance, be not well accounted for, and still less
their interchange with each other, when a ray of light is made
to pass through a second piece of spar situated transversely to
the first, yet the oblique refraction, when considered alone, seems
nearly as well explained as any other optical phenomenon."
Meanwhile the advocates of the corpuscular theory were not
idle ; and in the next few years a succession of discoveries on
their part, both theoretical and experimental, seemed likely to
imperil the good position to which Young had advanced the
rival hypothesis.
The first of these was a dynamical explanation of the
refraction of the extraordinary ray in crystals, which was
published in 1808 by Laplace.f His method is an extension of
that by which Maupertuis had accounted for the refraction of
the ordinary ray, and which since Maupertuis' day had been so
developed that it was now possible to apply it to problems of
all degrees of complexity. Laplace assumes that the crystalline
medium acts on the light-corpuscles of the extraordinary ray so
as to modify their velocity, in a ratio which depends on the
inclination of the extraordinary ray to the axis of the crystal :
so that, in fact, the difference of the squares of the velocities of
the ordinary and extraordinary rays is proportional to the
square of the sine of the angle which the latter ray makes with
the axis. The principle of least action then leads to a law of
refraction identical with that found by Huygens' construction
* Phil. Trans., 1802, p. 381.
tMem. de PInst., 1809, p. 300: Journal de Physique, Jan., 1809; Mem. de
la Soc. d'Arcueil, ii.
110 The Luminiferous Medium,
with the spheroid ; just as Maupertuis' investigation led to a
law of refraction for the ordinary ray identical with that found
by Huygens' construction with the sphere.
The law of refraction for the extraordinary ray may also be
deduced from Fermat's principle of least time, provided that the
velocity is taken inversely proportional to that assumed in the
principle of least action ; and the velocity appropriate to
Fermat's principle agrees with that found by Huygens, being, in
iact, proportional to the radius of the spheroid. These results
are obvious extensions of those already obtained for ordinary
refraction.
Laplace's theory was promptly attacked by Young,* who
pointed out the improbability of such a system of forces as
would be required to impress the requisite change of velocity on
the light-corpuscles. If the aim of controversial matter is to
convince the contemporary world, Young's paper must be
counted unsuccessful ; but it permanently enriched science by
proposing a dynamical foundation for double refraction on the
principles of the wave-theory. " A solution," he says, " might
•be deduced upon the Huygenian principles, from the simplest
possible supposition, that of a medium more easily compressible
in one direction than in any direction perpendicular to it, as if it
consisted of an infinite number of parallel plates connected by
a substance somewhat less elastic. Such a structure of the
elementary atoms of the crystal may be understood by compar-
ing them to a block of wood or of mica. Mr. Chladni found that
the mere obliquity of the fibres of a rod of Scotch fir reduced
the velocity with which it transmitted sound in the proportion
of 4 to 5. It is therefore obvious that a block of such wood
-must transmit every impulse in spheroidal — that is, oval —
undulations ;• and it may also be demonstrated, as we shall
show at the conclusion of this article, that the spheroid will be
truly elliptical when the body consists either of plane and
parallel strata, or of equidistant fibres, supposing both to be
^extremely thin, and to be connected by a less highly elastic
* Quarterly Eeview, Nov., 1809 ; Young's Works, i, p. 220.
from Bradley to FremeL 111
substance ; the spheroid being in the former case oblate and in
the latter oblong." Young then proceeds to a formal proof
that "an impulse is propagated through every perpendicular
section of a lamellar elastic substance in the form of an elliptic
undulation." This must be regarded as the beginning of
the dynamical theory of light in crystals. It was confirmed
in a striking way not long afterwards by Brewster,* who found
that compression in one direction causes an isotropic transparent
solid to become doubly-refracting.
Meanwhile, in January, 1808, the French Academy had
proposed as the subject for the physical prize in 1810, " To
furnish a mathematical theory of double refraction, and to
confirm it by experiment." Among those who resolved to
compete was Etienne Louis Malus (b. 1775, d. 1812), a colonel
of engineers who had seen service with Napoleon's expedition
to Egypt. While conducting experiments towards the end of
1808 in a house in the Kue des Enfers in Paris, Malus happened
to analyse with a rhomb of Iceland spar the light of the setting
sun reflected from the window of the Luxembourg, and was
surprised to notice that the two images were of very different
intensities. Following up this observation, he found that light
which had been reflected from glass acquires thereby a modifi-
cation similar to that which Huygens had noticed in rays
which have experienced double refraction, and which Newton
had explained by supposing rays of light to have " sides." This
discovery appeared so important that without waiting for the
prize competition he communicated it to the Academy in
December, 1808, and published it in the following month.f
" I have found," he said, " that this singular disposition,
which has hitherto been regarded as one of the peculiar effects
of double refraction, can be completely impressed on the
luminous molecules by all transparent solids and liquids."
" For example, light reflected by the surface of water at an
* Phil. Trans., 1815, p. 60.
tNouveau Bulletin des Sciences, par la Soc. Philomatique. i (1809), p. 266;
Memoires de la Soc. d'Arcueil, ii (1809).
112 The Luminiferous Medium,
angle of 52°45' has all the characteristics of one of the beams
produced by the double refraction of Iceland spar, whose
principal section is parallel to the plane which passes through
the incident ray and the reflected ray. If we receive this
reflected ray on any doubly- refracting crystal, whose principal
section is parallel to the plane of reflexion, it will not be divided
into two beams as a ray of ordinary light would be, but will be
refracted according to the ordinary law."
After this Malus found that light which has been refracted
at the surface of any transparent substance likewise possesses
in some degree this property, to which he gave the name
polarization. The memoir* which he finally submitted to the
Academy, and which contains a rich store of experimental and
analytical work on double refraction, obtained the prize in 1810 ;
its immediate effect as regards the rival theories of the ultimate
nature of light was to encourage the adherents of the corpuscular
doctrine ; for it brought into greater prominence the phenomena
of polarization, of which the wave-theorists, still misled by the
analogy of light with sound, were unable to give any account.
The successful discoverer was elected to the Academy of
Sciences, and became a member of the celebrated club of Arcueil.f
But his health, which had been undermined by the Egyptian
campaign, now broke down completely : and he died, at the age
of thirty-six, in the following year.
The polarization of a reflected ray is in general incomplete —
i.e. the ray displays only imperfectly the properties of light
which has been polarized by double refraction ; but for one
particular angle of incidence, which depends on the reflecting
body, the polarization of the reflected ray is complete. Malus
measured with considerable accuracy the polarizing angles for
glass and water, and attempted to connect them with the other
optical constants of these substances, the refractive indices and
dispersive powers, but without success. The matter was
* Mem. presentes a 1'Inst. par divers Savans, ii (1811), p. 303.
t So called from the village near Paris where Laplace and Berthollet had
their country-houses, and where the meetings took place. The club consisted of
a dozen of the most celebrated scientific men in France.
from Bradley to Fresnel. 113
afterwards taken up by David Brewster (b. 1781, d. 1868), who
in 1815* showed that there is complete polarization by reflexion
when the reflected and refracted rays satisfy the condition of
being at right angles to each other.
Almost at the same time Brewster made another discovery
which profoundly affected the theory of double refraction. It
had till then been believed that double refraction is always
of the type occurring in Iceland spar, to which Huygens'
construction is applicable. Brewster now found this belief to be
erroneous, and showed that in a large class of crystals there are
two axes, instead of one, along which there is no double
refraction. Such crystals are called Uaxal, the simpler type to
which Iceland spar belongs being called uniaxal.
The wave-theory at this time was still encumbered with
difficulties. Diffraction was not satisfactorily explained ; for
polarization no explanation of any kind was forthcoming ; the
Huygenian construction appeared to require two different
luminiferous media within doubly refracting bodies ; and the
universality of that construction had been impugned by
Brewster's discovery of biaxal crystals.
The upholders of the emission theory, emboldened by the
success of Laplace's theory of double refraction, thought the
time ripe for their final triumph ; and as a step to this, in
March, 1817, they proposed Diffraction as the subject of the
Academy's prize for 1818. Their expectation was disappointed ;
and the successful memoir afforded the first of a series of
reverses by which, in the short space of seven years, the
corpuscular theory was completely overthrown.
The author was Augustin Fresnel (b. 1788, d. 1827), the
son of an architect, and himself a civil engineer in the
Government service in Normandy. During the brief dominance
of Napoleon after his escape from Elba in 1815, Fresnel fell into
trouble for having enlisted in the small army which attempted
to bar the exile's return ; and it was during a period of enforced
idleness following on his arrest that he commenced to study
•Phil. Trans., 1815, p. 125.
I
114 The Lumini/erous Medium,
diffraction. In his earliest memoir* he propounded a theory
similar to that of Young, which was spoiled like Young's
theory by the assumption that the fringes depend on light
reflected by the diffracting edge. Observing, however, that the
blunt and sharp edges of a knife produce exactly the same
fringes, he became dissatisfied with this attempt, and on July
15th, 1816, presented to the Academy a supplement to his
paper,f in which, for the first time, diffraction-effects are
referred to their true cause — namely, the mutual interference
of the secondary waves emitted by those portions of the original
wave-front which have not been obstructed by the diffracting
screen. Fresnel's method of calculation utilized the principles
of both Huygens and Young ; he summed the effects due to
different portions of the same primary wave-front, with due
regard to the differences of phase engendered in propagation.
The sketch presented to the Academy in 1816 was during
the next two years developed into an exhaustive memoir, J
which was submitted for the Academy's prize.
It so happened that the earliest memoir, which had been
presented to the Academy in the autumn of 1815, had been
referred to a Commission of which the reporter was Francois
Arago (&. 1786, d. 1853) ; Arago was so much impressed that
he sought the friendship of the author, of whom he was later a
strenuous champion.
A champion was indeed needed when the larger memoir was
submitted ; for Laplace, Poisson, and Biot, who constituted a
majority of the Commission to which it was referred, were all
zealous supporters of the corpuscular theory. During the
examination, however, Fresnel was vindicated in a somewhat
curious way. He had calculated in the memoir the diffraction-
patterns of a straight edge, of a narrow opaque body bounded
by parallel sides, and of a narrow opening bounded by parallel
edges, and had shown that the results agreed excellently with
* Annales de Chimie (2), i (1816), p 239 ; (Euvres, i, p. 89.
t (Euvres, i, p. 129.
I Mem. de 1'Acad., v (1826), p. 339 ; (Euvres, i, p. 247.
from Bradley to FresneL 115
his experimental measures. Poisson, when reading the manu-
script, happened to notice that the analysis could be extended
to other cases, and in particular that it would indicate the
existence of a bright spot at the centre of the shadow of a
circular screen. He suggested to Fresnel that this and some
further consequences should be tested experimentally ; this was
done, and the results were found to confirm the new theory.
The concordance of observation and calculation was so admirable
in all cases where a comparison was possible that the prize was
awarded to Fresnel without further hesitation.
In the same year in which the memoir on diffraction was
submitted, Fresnel published an investigation* of the influence
of the earth's motion on light. We have already seen that
aberration was explained by its discoverer in terms of the
corpuscular theory ; and it was Young who first showedf how
it may be explained on the wave-hypothesis. " Upon con-
sidering the phenomena of the aberration of the stars," he
wrote, " I am disposed to believe that the luminiferous aether
pervades the substance of all material bodies with little or no
resistance, as freely perhaps as the wind passes through a
grove of trees." In fact, if we suppose the aether surrounding
the earth to be at rest and unaffected by the earth's motion,
the light- waves will not partake of the motion of the telescope ,
which we may suppose directed to the true place of the star,
and the image of the star will therefore be displaced from the
central spider-line at the focus by a distance equal to that
which the earth describes while the light is travelling through
the telescope. This agrees with what is actually observed.
But a host of further questions now suggest themselves.
Suppose, for instance, that a slab of glass with a plane face is
carried along by the motion of the earth, and it is desired to
adjust it so that a ray of light coming from a certain star
shall not be bent when it enters the glass : must the
.surface be placed at right angles to the true direction of the
* Annales de Chimie, ix, p. 57 (1818) ; CEnvres, ii, p. 627.
t Phil. Trans., 1804, p. 12; Young's Works, i, p. 188.
I 2
116 The L uminiferous Medium,
star as freed from aberration, or to its apparent direction as
affected by aberration ? The question whether rays coming
from the stars are refracted differently from rays origi-
nating in terrestrial sources had been raised originally by
Michell* ; and Kobison and Wilsonf had asserted that the focal
length of an achromatic telescope should be increased when it
is directed to a star towards which the earth is moving, owing
to the change in the relative velocity of light. AragoJ sub-
mitted the matter to the test of experiment, and concluded that
the light coming from any star behaves in all cases of reflexion
and refraction precisely as it would if the star were situated in
the place which it appears to occupy in consequence of aber-
ration, and the earth were at rest ; so that the apparent
refraction in a moving prism is equal to the absolute refraction
in a fixed prism.
Fresnel now set out to provide a theory capable of explaining
Arago's result. To this end he adopted Young's suggestion,
that the refractive powers of transparent bodies depend on the
concentration of aether within them ; and made it more precise
by assuming that the aethereal density in any body is pro-
portional to the square of the refractive index. Thus, if c
denote the velocity of light in vacuo, and if c, denote its
velocity in a given material body at rest, so that /u = c/o{ is the
refractive index, then the densities p and pl of the aether in
interplanetary space and in the body respectively will be
connected by the relation
pi = n*P-
Fresnel further assumed that, when a body is in motion, part
of the aether within it is carried along — namely, that part which
constitutes tne excess of its density over the density of aether
in vacuo ; while the rest of the aether within the space occupied
by the body is stationary. Thus the density of aether carried
* Phil. Trans., 1784, p. 35.
t Trans. E. S. Edin., i, Hist., p. 30.
J Biot, Astron. Phys., 3rd ed., v, p. 364. The accuracy of Arago's
experiment can scarcely have been such as to demonstrate absolutely his
result.
ft,
from Bradley to FresneL 117
along is (pi - p) or (^ - l)/o, while a quantity of aether of
density p remains at rest. The velocity with which the centre
of gravity of the aether within the body moves forward in the
direction of propagation is therefore
where w denotes the component of the velocity of the body in
this direction. This is to be added to the velocity of propaga-
tion of the light- waves within the body ; so that in the moving
body the absolute velocity of light is
Many years afterwards Stokes* put the same supposition in
a slightly different form. Suppose the whole of the aether
within the body to move together, the aether entering the body
in front, and being immediately condensed, and issuing from it
behind, where it is immediately rarefied. On this assumption a
mass pw of aether must pass in unit time across a plane of area
unity, drawn anywhere within the body in a direction at right
angles to the body's motion; and therefore the aether within
the body has a drift- velocity - wp/pl relative to the body : so
the velocity of light relative to the body will be Ci - wplp\, and
the absolute velocity of light in the moving body will be
v*
v* k
or ci + £^i w, as before.
P
This formula was experimentally confirmed in 1851 by
H. Fizeau,f who measured the displacement of interference-
fringes formed by light which had passed through a column of
moving water.
* Phil. Mag. xxviii (1846) p. 76.
t Annales de Chimie, Ivii (1859), p. 385. Also by A. A. Michelson and
E. W. Morley, Am. Journ. Science, xxxi (1886), p. 377.
118 The Luminiferous Medium,
The same result may easily be deduced from an experiment
performed by Hoek.* In this a beam of light was divided into
two portions, one of which was made to pass
through a tube of water AB and was then reflected
at a mirror C, the light being afterwards allowed to
return to A without passing through the water :
while the other portion of the bifurcated beam was
made to describe the same path in the reverse
order, i.e. passing through the water on its return
,. journey from C instead of on the outward journey,
On causing the two portions of the beam to inter-
fere, Hoek found that no difference of phase was produced
between them when the apparatus was oriented in the direction
of the terrestrial motion.
Let w denote the velocity of the earth, supposed to be
directed from the tube towards the mirror. Let c/n denote the
velocity of light in the water at rest, and C/A* + <l> the velocity
of light in the water when moving. Let I denote the length of
the tube. The magnitude of the distance BC does not affect
the experiment, so we may suppose it zero.
The time taken by the first portion of the beam to perform
its journey is evidently
If i
C/fi + ^ — W C + W '
while the time for the second portion of the beam is
I I
+ —.
C - W C/fJL - 0 + W
The equality of these expressions gives at once, when terms
of higher orders than the first in w/c are neglected,
0 = Ou2 - 1) w\^\
which is FresneFs expression.!
* Archives Neerl. iii, 180 (1868).
t Fresnel's law may also be deduced from the principle that the amount of light
transmitted by a slab of transparent matter must be the same whether the slab is at
rest or in motion : otherwise the equilibrium of exchanges of radiation would be
tiated. Cf. Larmor, Phil. Trans, clxxxv (1893), p, 775.
from Bradley to Fresnel.
119
On the basis of this formula, Fresnel proceeded to solve
the problem of refraction in moving bodies. Suppose that a
prism AQ (70 B0 is carried along by the earth's motion in vacuo, its
face A(, C0 being at right angles to the direction of motion ; and
that light from a star is incident normally on this face. The
rays experience no refraction at incidence ; and we have only to
consider the effect produced by the second surface A<>I>0. Sup-
pose that during an interval T of time the prism travels from
the position AQ C0 Bo to the position A± Ci B^ while the luminous
disturbance at C0 travels to £h and the luminous disturbance at
A0 travels to D, so that Bv D is the emergent wave-front.
Then we have
-1
10
A0D
TC,
If we write CiA\B\ = i, and denote the total deviation
of the wave-front by 81, we have
AiD = AJ) - A±AQ cos Si = TC - rw cos 81,
TIC,
120 The Lumniiferous Medium,
and therefore (neglecting second-order terms in w/c]
A
sin A^B^D c - w cos 81 _ c w w ^
-— — • : ~ — ~ — "f- — •*" COo Ol«
sin ^ Ci ct c Ci
Ct-W-
Denoting by 8 the value of 81 when w is zero, we have
sin (i -8) c
sin i d
Subtracting this equation from the preceding, we have
8 -Si _ w
sin § c
Now the telescope by which the emergent wave-front B\ D
is received is itself being carried forward by the earth's motion;
and we must therefore apply the usual correction for aberration
in order to find the apparent direction of the emergent ray. But
this correction is w sin 8/c, and precisely counteracts the effect
which has been calculated as due to the motion of the prism.
So finally we see that the motion of the earth has no first-order
influence on the refraction of light from the stars.
Fresnel inferred from his formula that if observations were
made with a telescope filled with water, the aberration would be
unaffected by the presence of the water — a result which was
verified by Airy* in 1871. He showed, moreover, that the
apparent positions of terrestrial objects, carried along with the
observer, are not displaced by the earth's motion ; that experi-
ments in refraction and interference are not influenced by any
motion which is common to the source, apparatus, and observer ;
and that light travels between given points of a moving material
system by the path of least time. These predictions have also been
confirmed by observation: Kespighif in 1861, and Hoek+ in 1868,
experimenting with a telescope filled with water and a terrestrial
source of light, found that no effect was produced on the
phenomena of reflexion and refraction by altering the orienta-
* Proc. Roy. Soc., xx, p. 35. t Mem. Accad. Sci. Bologna, ii, p. 279.
I Ast. Nach., Ixxiii, p. 193.
from Bradtey to Fresnel. 121
tion of the apparatus relative to the direction of the earth's
motion. E. Mascart* in 1872 discussed experimentally the
question of the effect of motion of the source or recipient of
light in all its bearings, and showed that the light of the sun and
that derived from artificial sources are alike incapable of revealing
by diffraction-phenomena the translatory motion of the earth.
The greatest problem now confronting the investigators of
light was to reconcile the facts of polarization with the principles
of the wave-theory. Young had long been pondering over this,
but had hitherto been baffled by it. In 1816 he received a
visit from Arago, who told him of a new experimental result
which he and Fresnel had lately obtained! — namely, that two
pencils of light, polarized in planes at right angles, do not
interfere with each other under circumstances in which ordinary
light shows interference-phenomena, but always give by their
reunion the same intensity of light, whatever be their difference
of path.
Arago had not long left him when Young, reflecting on the
new experiment, discovered the long-sought key to the mystery :
it consisted in the very alternative which Bernoulli had rejected
eighty years before, of supposing that the vibrations of light are
executed at right angles to the direction of propagation.
Young's ideas were first embodied in a letter to Arago,J
dated Jan. 12, 1817. "I have been reflecting," he wrote, " on the
possibility of giving an imperfect explanation of the affection
of light which constitutes polarization, without departing from
the genuine doctrine of undulations. It is a principle in this
theory, that all undulations are simply propagated through
homogeneous mediums in concentric spherical surfaces like the
•Ann. de 1'Ecole Noemale, (2) i, p. 157.
t It was not published until 1819, in Annales de Chimie, x ; Fresnel's (Euvres,
i, p. 509. By means of this result, Fresnel was able to give a complete explana-
tion of a class of phenomena which Arago had discovered in 1811, viz. that when
polarized light is transmitted through thin plates of sulphate of lime or mica, and
afterwards analysed by a prism of Iceland spar, beautiful complementary colours
are displayed. Young had shown that these effects are due essentially to inter-
ference, hut had not made clear the part played by polarization.
J Young's JTorks, i., p. 380.
122 The Lumini/erous Medium,
undulations of sound, consisting simply in the direct and retro-
grade motions of the particles in the direction of the radius,,
with their concomitant condensation and rarefactions. And
yet it is possible to explain in this theory a transverse vibration,,
propagated also in the direction of the radius, and with equal
velocity, the motions of the particles being in a certain constant
direction with respect to that radius ; and this is a polarization"
In an article on " Chromatics," which was written in
September of the same year* for the supplement to the
Encyclopaedia Britannica, he says :f " If we assume as a mathe-
matical postulate, on the undulating theory, without attempting
to demonstrate its physical foundation, that a transverse motion
may be propagated in a direct line, we may derive from this
assumption a tolerable illustration of the subdivision of polarized
light by reflexion in an oblique plane," by " supposing the polar
motion to be resolved " into two constituents, which fare
differently at reflexion.
In a further letter to Arago, dated April 29th, 1818, Young
recurred to the subject of transverse vibrations, comparing light
to the undulations of a cord agitated by one of its extremities.^
This letter was shown by Arago to Fresnel, who at once saw
that it presented the true explanation of the non-interference
of beams polarized in perpendicular planes, and that the latter
effect could even be made the basis of a proof of the correctness
of Young's hypothesis : for if the vibration of each beam be
supposed resolved into three components, one along the ray and
the other two at right angles to it, it is obvious from the Arago-
Fresnel experiment that the components in the direction of the
ray must vanish : in other words, that the vibrations which
constitute light are executed in the wave-front.
It must be remembered that the theory of the propagation
of waves in an elastic solid was as yet unknown, and light was
* Peacock's Life of Young, p. 391. t Young's Works, i., p. 279.
JThis analogy had been given by Hooke in a communication to the Royal
Society on Feb. 15, 1671-2. But there seems no reason to suppose that Hook e-
appreciated the point now advanced by Young.
from Bradley to FresneL 123
still always interpreted by the analogy with the vibrations of
sound in air, for which the direction of vibration is the same as
that of propagation. It was therefore necessary to give some
justification for the new departure. With wonderful insight
Fresnel indicated* the precise direction in which the theory of
vibrations in ponderable bodies needed to be extended in order
to allow of waves similar to those of light : " the geometers," he
wrote, " who have discussed the vibrations of elastic fluids hitherto
have taken account of no accelerating forces except those arising
from the difference of condensation or dilatation between conse-
cutive layers." He pointed out that if we also suppose the
medium to possess a rigidity, or power of resisting distortion, such
as is manifested by all actual solid bodies, it will be capable of
transverse vibration. The absence of longitudinal waves in the
aether he accounted for by supposing that the forces which oppose
condensation are far more powerful than those which oppose
distortion, and that the velocity with which condensations are
propagated is so great compared with the speed of the oscillations
of light, that a practical equilibrium of pressure is maintained
perpetually.
The nature of ordinary non-polarized light was next discussed.
" If then," Fresnel wrote,f " the polarization of a ray of light
consists in this, that all its vibrations are executed in the same
direction, it results from any hypothesis on the generation of
light-waves, that a ray emanating from a single centre of dis-
turbance will always be polarized in a definite plane at any
instant. But an instant afterwards, the direction of the motion
changes, and with it the plane of polarization ; and these
variations follow each other as quickly as the perturbations of
the vibrations of the luminous particle : so that even if we could
*Annales de Chiinie, xvii (1821), p. 180; (Eiwres, i, p. 629. Young had
already drawn attention to this point. " It is difficult," he says in his Lectures on
Natural Philosophy, ed. 1807, vol. i, p. 138, "to compare the lateral adhesion, or
the force which resists the detrusion of the parts of a solid, with any form of direct
cohesion. This force constitutes the rigidity or hardness of a solid body, and is
wholly absent from liquids."
t Loc. cit, p. 185.
124 The Luminiferous Medium,
isolate the light of this particular particle from that of other
luminous particles, we should doubtless not recognize in it any
appearance of polarization. If we consider now the effect pro-
duced by the union of all the waves which emanate from the
different points of a luminous body, we see that at each instant,
at a definite point of the aether, the general resultant of all the
motions which commingle there will have a determinate
direction, but this direction will vary from one instant to the
next. So direct light can be considered as the union, or more
exactly as the rapid succession, of systems of waves polarized in
all directions. According to this way of looking at the matter,
the act of polarization consists not in creating these transverse
motions, but in decomposing them in two invariable directions,
and separating the components from each other ; for . then, in
each of them, the oscillatory motions take place always in the
same plane."
He then proceeded to consider the relation of the direction of
vibration to the plane of polarization. " Apply these ideas to
double refraction, and regard a uniaxal crystal as an elastic
medium in which the accelerating force which results from
the displacement of a row of molecules perpendicular to the
axis, relative to contiguous rows, is the same all round the
axis ; while the displacements parallel to the axis produce
accelerating forces of a different intensity, stronger if the
crystal is "repulsive," and weaker if it is "attractive." The
distinctive character of the rays which are ordinarily refracted
being that of propagating themselves with the same velocity
in all directions, we must admit that their oscillatory motions
are executed at right angles to the plane drawn through these
rays and the axis of the crystal; for then the displacements
which they occasion, always taking place along directions
perpendicular to the axis, will, by hypothesis, always give rise
to the same accelerating forces. But, with the conventional
meaning which is attached to the expression 'plane, of polarization,
the plane of polarization of the ordinary rays is the plane
through the axis : thus, in a pencil of polarized light, the
from Bradley, to Fresnel. 125
oscillatory motion is executed at right angles to the plane of
polarization"
This result afforded Fresnel a foothold in dealing with the
problem which occupied the rest of his life : henceforth his aim
was to base the theory of light on the dynamical properties of
the luminiferous medium.
The first topic which he attacked from this point of view
was the propagation of light in crystalline bodies. Since
Brewster's discovery that many crystals do not conform to the
type to which Huygens' construction is applicable, the wave
theory had to some extent lost credit in this region. Fresnel,
now, by what was perhaps the most brilliant of all his efforts,*
not only reconquered the lost territory, but added a new domain
to science.
He had, as he tells us himself, never believed the doctrine
that in crystals there are two different luminiferous media,
one to transmit the ordinary, and the other the extraordinary
waves. The alternative to which he inclined was that the two
velocities of propagation were really the two roots of a quadratic
equation, derivable in some way from the theory of a single
aether. Could this equation be obtained, he was confident of
finding the explanation, not only of double refraction, but also
of the polarization by which it is always accompanied.
The first step was to take the case of uniaxal crystals,
which had been discussed by Huygens, and to see whether
Huygens' sphere and spheroid could be replaced by, or made to
depend on, a single surface.f
Now a wave propagated in any direction through a uniaxal
*His first memoir on Double Refraction was presented to the Academy on
Nov. 19th, 1821, but has not been published except in his collected works:
(Eitvres, ii, p. 261. It was followed by other papers in 1822; and the results were
finally collected in a memoir which was printed in 1827, Mem. de VAcad. vii,
p. 45, (Euvres, ii, p. 479.
t In attempting to reconstruct Fresnel's course of thought at this period, the
present writer has derived much help from the Life prefixed to the (Euvres de
Fresnel. Both Fresnel and Young were singularly fortunate in their biographers :
Peacock's Life of Young, and this notice of Fresnel, which was the last work of
Verdet, are excellent reading.
126 The Luminiferous Medium,
crystal can be resolved into two plane-polarized components ;
one of these, the " ordinary ray," is polarized in the principal
section, and has a velocity vl9 which may be represented by the
radius of Huygens' sphere — say,
Vi = &;
while the other, the " extraordinary ray," is polarized in a plane
.at right angles to the principal section, and has a wave- velocity v9,
which may be represented by the perpendicular drawn from the
centre of Huygens' spheroid on the tangent-plane parallel to the
plane of the wave. If the spheroid be represented by the
equation
if + z'" x*
— + ^ = 1-
and if (I, m, n) denote the direction-cosines of the normal to the
plane of the wave, we have therefore
v,~ = a*(m* + n*) + ?>2/2.
But the quantities 1/Vi and l/t?8, as given by these equations,
are easily seen to be the lengths of the semi-axes of the ellipse
in which the spheroid
62(?/3 4- z~) + arx- = 1
is intersected by the plane
Ix + my + nz = 0 ;
.and thus the construction in terms of Huygens' sphere and
spheroid can be replaced by one which depends only on a single
surface, namely the spheroid
Having achieved this reduction, Fresnel guessed that the
<?ase of biaxal crystals could be covered by substituting for the
latter spheroid an ellipsoid with three unequal axes — say,
xz if z*
_+£+_ =
If I/Vi and l/^ denote the lengths of the semi-axes of the
.ellipse in which this ellipsoid is intersected by the plane
Ix 4 my + nz - 0,
from Bradley to Fresnel. 127
it is well known that #1 and vz are the roots of the equation in v
; --0;
1 . 1 ,1
tf tf v-
ti £2 «3
and accordingly Fresnel conjectured that the roots of this
equation represent the velocities, in a biaxal crystal, of the two
plane-polarized waves whose normals are in the direction
(I, m, n).
Having thus arrived at his result by reasoning of a purely
geometrical character, he now devised a dynamical scheme to
suit it.
The vibrating medium within a crystal he supposed to be
ultimately constituted of particles subjected to mutual forces ;
and on this assumption he showed that the elastic force of
restitution when the system is disturbed must depend linearly
on the displacement. In this first proposition a difference is
apparent between Fresnel's and a true elastic-solid theory ; for
in actual elastic solids the forces of restitution depend not on
the absolute displacement, but on the strains, i.e., the relative
displacements.
In any crystal there will exist three directions at right
angles to each other, for which the force of restitution acts in
the same line as the displacement : the directions which possess
this property are named axes of elasticity. Let these be taken
as axes, and suppose that the elastic forces of restitution for
unit displacements in these three directions are 1/5], l/c2, l/«s
respectively. That the elasticity should vary with the direction
of the molecular displacement seemed to Fresnel to suggest that
the molecules of the material body either take part in the
luminous vibration, or at any rate influence in some way the
elasticity of the aether.
A unit displacement in any arbitrary* direction (a, )3, 7) can
be resolved into component displacements (cos a, cos /3, cos 7)
parallel to the axes, and each of these produces its own effect
128 The Luminiferous Medium ^
independently ; so the components of the force of restitution are
COS a COS )3 COS y
€l ft £3
This resultant force has not in general the same direction
as the displacement which produced it ; but it may always
he decomposed into two other forces, one parallel and the other
perpendicular to the direction of the displacement ; and the
former of these is evidently
The surface
COS2 a COS2 )3 COS2 7
I {_ I £_
fl €2 £3
X2 V*
£i £2 £3
will therefore have the property that the square of its radius
vector in any direction is proportional to the component in that
direction of the elastic force due to a unit displacement in that
direction : it is called the surface of elasticity.
Consider now a displacement along one of the axes of the
section on which the surface of elasticity is intersected by the
plane of the wave. It is easily seen that in this case the com-
ponent of the elastic force at right angles to the displacement
acts along the normal to the wave-front; and Fresnel assumes
that it will be without influence on the propagation of the
vibrations, on the ground of his fundamental hypothesis that the
vibrations of light are performed solely in the wave-front. This
step is evidently open to criticism ; for in a dynamical theory
everything should be deduced from the laws of motion without
special assumptions. But granting his contention, it follows
that such a displacement will retain its direction, and will be
propagated as a plane-polarized wave with a definite velocity.
Now, in order that a stretched cord may vibrate with
unchanged period, when its tension is varied, its length must be
increased proportionally to the square root of its tension ; and
similarly the wave-length of a luminous vibration of given period
is proportional to the square root of the elastic force (per unit
from Bradley to Fresnel. 129
displacement), which urges the molecules of the medium parallel
to the wave-front. Hence the velocity of propagation of a
wave, measured at right angles to its front, is proportional to
the square root of the component, along the direction of dis-
placement, of the elastic force per unit displacement ; and the
velocity of propagation of such a plane-polarized wave as we
have considered is proportional to the radius vector of the
surface of elasticity in the direction of displacement.
Moreover, any displacement in the given wave-front can be
resolved into two, which are respectively parallel to the two
axes of the diametral section of the surface of elasticity by a
plane parallel to this wave-front ; and it follows from what has
been said that each of these component displacements will be
propagated as an independent plane-polarized wave, the velocities
of propagation being proportional to the axes of the section,*
and therefore inversely proportional to the axes of the section of
the inverse surface of this with respect to the origin, which is
the ellipsoid
* + £ + *-i.
£i £2 £3
But this is precisely the result to which, as we have seen,
Fresnel had been led by purely geometrical considerations ; and
thus his geometrical conjecture could now be regarded as
substantiated by a study of the dynamics of the medium.
It is easy to determine the wave-surface or locus at any
instant —say, t = 1 — of a disturbance originated at some previous
instant — say,£ = 0 — at some particular point — say, the origin. For
this wave-surface will evidently be the envelope of plane waves
emitted from the origin at the instant t = 0 — that is, it will be
the envelope of planes
Ix + my + nz - v = 0,
where the constants /, m, n, v are connected by the identical
equation I2 + m* + nz = 1,
* It is evident from this that the optic axes, or lines of single wave-velocity,
along which there is no double refraction, will be perpendicular to the two
circular sections of the surface of elasticity.
K
130 The Lumimferous Medium,
and by the relation previously found — namely,
/2 m2 n~
1
By the usual procedure for determining envelopes, it may be
shown that the locus in question is the surface of the fourth
degree
xz_ _f _fl_ _ n
which is called Fresnel's wave-surface* It is a two-sheeted surface,
as must evidently be the case from physical considerations. In
uniaxal crystals, for which *2 and c 3 are equal, it degenerates into
the sphere
r2 = l/e>,
and the spheroid
^ + fl (tf + Z2) = 1.
It is to these two surfaces that tangent-planes are drawn in
the construction given by Huygens for the ordinary and
extraordinary refracted rays in Iceland spar. As Fresnel
observed, exactly the same construction applies to biaxal
crystals, when the two sheets of the wave-surface are substi-
tuted for Huygens' sphere and spheroid.
" The theory which I have adopted," says Fresnel at the end
of this memorable paper, " and the simple constructions which
I have deduced from it, have this remarkable character, that
all the unknown quantities are determined together by the
solution of the problem. We find at the same time the
velocities of the ordinary ray and of the extraordinary ray, and
their planes of polarization. Physicists who have studied
attentively the laws of nature will feel that such simplicity and
* Another construction for the wave-surface is the following, which is due to
MacCullagh, Coll. Works, p. 1. Let the ellipsoid
*ix~ + 62^" ~*~ *3~~ = *
be intersected hy a plane through its centre, and on the perpendicular to that plane
take lengths equal to the semi-axes of the section. The locus of these extremities
is the wave-surface.
from Bradley to FresneL 131
such close relations between the different elements of the
phenomenon are conclusive in favour of the hypothesis on
which they are based."
The question as to the correctness of Fresnel's construction
was discussed for many years afterwards. A striking conse-
quence of it was pointed out in 1832 by William Kowan
Hamilton (b. 1805, d. 1865), Royal Astronomer of Ireland, who
remarked* that the surface defined by Fresnel's equation has
four conical points, at each of which there is an infinite number
of tangent planes ; consequently, a single ray, proceeding from
a point within the crystal in the direction of one of these
points, must be divided on emergence into an infinite number of
rays, constituting a conical surface. Hamilton also showed
that there are four planes, each of which touches the wave-
surface in an infinite number of points, constituting a circle of
contact : so that a corresponding ray incident externally should
be divided within the crystal into an infinite number of refracted
rays, again constituting a conical surface.
These singular and unexpected consequences of the theory
were shortly afterwards verified experimentally by Humphrey
Lloyd,f and helped greatly to confirm belief in Fresnel's theory.
It should, however, be observed that conical refraction only
shows his form of the wave- surf ace to be correct in its general
features, and is no test of its accuracy in all details. But it
was shown experimentally by Stokes in 1872J Glazebrook in
1879,§ and Hastings in 1887,1 1 that the construction of Huygens
and Fresnel is certainly correct to a very high degree of
approximation; and Fresnel's final formulae have since been
regarded as unassailable. The dynamical substructure on
which he based them is, as we have seen, open to objection ;
* Trans. Roy. Irish Acad., xvii (1833), p. 1.
t Trans. Roy. Irish Acad., xvii (1833), p. 145. Strictly speaking, the bright
oone which is usually observed arises from rays adjacent to the singular ray :
the latter can, however, be observed, its enfeeblement by dispersion into the
conical form causing it to appear dark.
I Proc. R. S., xx, p. 443.
§ Phil. Trans., clxxi, p. 421.
|| Am. Jour. Sci. (3), xxxv, p. 60.
K 2
132 The Luminiferous Medium,
but, as Stokes observed*: "If we reflect on the state of the
subject as Fresnel found it, and as he left it, the wonder is, not
that he failed to give a rigorous dynamical theory, but that a
single mind was capable of effecting so much."
In a second supplement to his first memoir on Double
Eefraction, presented to the Academy on November 26th, 1821,-]-
Fresnel indicated the lines on which his theory might be
extended so as to take account of dispersion. " The molecular
groups, or the particles of bodies," he wrote, " may be separated
by intervals which, though small, are certainly not altogether
insensible relatively to the length of a wave." Such a coarse-
grainedness of the medium would, as he foresaw, introduce into
the equations terms by which dispersion might be explained ;
indeed, the theory of dispersion which was afterwards given by
Cauchy was actually based on this principle. It seems likely
that, towards the close of his life, Fresnel was contemplating a
great memoir on dispersion^ which was never completed.
Fresnel had reason at first to be pleased with the reception of
his work on the optics of crystals : for in August, 1822, Laplace
spoke highly of it in public ; and when at the end of the year a
seat in the Academy became vacant, he was encouraged to hope
that the choice would fall on him. In this he was disappointed. §.
Meanwhile his researches were steadily continued ; and in
January, 1823, the very month of his rejection, he presented to-
the Academy a theory in which reflexion and refraction] | are
referred to the dynamical properties of the luminiferous media.
*Brit. Assoc. Rep., 1862, p. 254.
t (Euvres, ii, p. 438.
J Cf. the biography in (Euvres de Fresnel, i, p. xcvi.
§ Writing to Young in the spring of 1823, he says : " Tous ces memoires,
que dernierement j'ai pre'sentes coup sur coup a 1'Academie des Sciences, ne
m'en ont pas cependant otivert la porte. C'est M. Dulong qui a ete nomine
pour remplir la place vacante dans la section de physique. . . Vous voyez,
Monsieur, que la theorie des ondulations ne m'a point porte honheur : mais cela
ne m'en degoute pas : et je me console de ce malheur en m* occupant d'optique
avec une nouvelle ardeur."
|| The MSS- was for some time believed to be lost, but was ultimately found
among the papers of Fourier, and printed in Mem. de 1'Acad. xi (1832), p. 393 :
(Euvres, i, p. 767.
from Bradley to FresneL 133
As in his previous investigations, he assumes that the
vibrations which constitute light are executed at right angles
to the plane of polarization. He adopts Young's principle, that
reflexion and refraction are due to differences in the inertia of
the aether in different material bodies, and supposes (as in
his memoir on Aberration) that the inertia is proportional to
the inverse square of the velocity of propagation of light in
the medium. The conditions which he proposes to satisfy at the
interface between two media are that the displacements of the
adjacent molecules, resolved parallel to this interface, shall be
equal in the two media ; and that the energy of the reflected
and refracted waves together shall be equal to that of the
incident wave.
On these assumptions the intensity of the reflected and
refracted light may be obtained in the following way : —
Consider first the case in which the incident light is
polarized in the plane of incidence, so that the displacement is
at right angles to the plane of incidence ; let the amplitude
of the displacement at a given point of the interface be /
for the incident ray, g for the reflected ray, and h for the
refracted ray.
The quantities of energy propagated per second across unit
cross-section of the incident, reflected, and refracted beams are
proportional respectively to
where cb c2, denote the velocities of light, and pl} pz the densities
of aether, in the two media ; and the cross-sections of the beams
which meet the interface in unit area are
cos i, cos i, cos r
respectively. The principle of conservation of energy therefore
gives
c,p! cos i ./2 = c,/o! cos i . gz + c2/o2 cos r . h~.
The equation of continuity of displacement at the interface is
/ + 9 = h.
134 The Luminiferous Medium,
Eliminating li between these two equations, and using the
formulae
sin2 T Co2 pi
sin2 i C* p2 '
we obtain the equation
Z. _ sm (^ ~ r)
g sin (i + r)
Thus when the light is polarized in the plane of reflexion, the
amplitude of the reflected wave is
Q-l -T\ (ft ^ -0*\
- — \-. r x the amplitude of the incident vibration.
sin pj + r)
Fresnel shows in a similar way that when the light is
polarized at right angles to the plane of reflexion, the ratio of
the amplitudes of the reflected and incident waves is
tan (i - r)
tan (i + r)
These formulae are generally known as Fresnel' s sine-law and
FresneTs tangent-law respectively. They had, however, been
discovered experimentally by Brewster some years previously.
When the incidence is perpendicular, so that i and r are very
small, the ratio of the amplitudes becomes
Limit ,
^ + r
or
where ju2 and //i denote the refractive indices of the media.
This formula had been given previously by Young* and Poisson,f
on the supposition that the elasticity of the aether is of the
same kind as that of air in sound.
When i + r = 90°, tan (i + r) becomes infinite : and thus
a theoretical explanation is obtained for Brewster 's law, that if
the incidence is such as to make the reflected and refracted rays-
* Article Chromatics, Encycl. Britt. Suppl. t Mem. Inst. ii. (1817).
fro vi Bradley to Fresnel. 135
perpendicular to each other, the reflected light will be wholly
polarized in the plane of reflexion.
Fre&nel's investigation can scarcely be called a dynamical
theory in the strict sense, as the qualities of the medium are
not defined. His method was to work backwards from the
known properties of light, in the hope of arriving at a mechanism
to which they could be attributed ; he succeeded in accounting
for the phenomena in terms of a few simple principles, but was
not able to specify an aether which would in turn account for
these principles. The " displacement " of Fresnel could not be
a displacement in an elastic solid of the usual type, since its
normal component is not continuous across the interface between
two media.*
The theory of ordinary reflexion was completed by a dis-
cussion of the case in which light is reflected totally. This had
formed the subject of some of Fresnel's experimental researches
several years before; and in two papersf presented to the
Academy in November, 1817, and January, 1818, he had shown
that light polarized in any plane inclined to the plane of reflexion
is partly "depolarized" by total reflexion, and that this is
due to differences of phase which are introduced between the
components polarized in and perpendicular to the plane of
reflexion. " When the reflexion is total," he said, " rays
polarized in the plane of reflexion are reflected nearer the
surface of the glass than those polarized at right angles to the
same plane, so that there is a difference in the paths described."
This change of phase he now deduced from the formulae
already obtained for ordinary reflexion. Considering light
polarized in the plane of reflexion, the ratio of the amplitudes of
the reflected and incident light is, as we have seen,
sin (i - r)
sin (i + r) '
when the sine of the angle of incidence is greater than /i2/jui,
* Fresnel's theory of reflexion can, however, he reconciled with the electro-
magnetic theory of light, by identifying his "displacement" with the electric
force. f (Euvres de Fresnel, i., pp. 441, 487.
136 The Luminiferous Medium.
so that total reflexion takes place, this ratio may be written in
the form
where 6 denotes a real quantity defined by the equation
tan
cos ^
Fresnel interpreted this expression to mean that the
amplitude of the reflected light is equal to that of the incident,
but that the two waves differ in phase by an amount 0. The
case of light polarized at right angles to the plane of reflexion
may be treated in the same way, and the resulting formulae are
completely confirmed by experiment.
A few months after the memoir on reflexion had been
presented, Fresnel was elected to a seat in the Academy ; and
during the rest of his short life honours came to him both from
France and abroad. In 1827 the Royal Society awarded him
the Rumford medal ; but Arago, to whom Young had confided
the mission of conveying the medal, found him dying ; and
eight days afterwards he breathed his last.
By the genius of Young and Fresnel the wave-theory of
light was established in a position which has since remained
unquestioned ; and it seemed almost a work of supererogation
when, in 1850, Foucault* and Fizeau,f carrying out a plan long
before imagined by Arago, directly measured the velocity of
light in air and in water, and found that on the question so
long debated between the rival schools the adherents of the
undulatory theory had been in the right.
* Comptes Rendus, xxx (1850), p. 551. t Ibid., p. 562.
( 137 )
CHAPTER V.
THE AETHER AS AN ELASTIC SOLID.
WHEN Young and Fresnel put forward the view that the
vibrations of light are performed at right angles to its direction
of propagation, they at the same time pointed out that this
peculiarity might be explained by making a new hypothesis
regarding the nature of the luminiferous medium ; namely, that
it possesses the power of resisting attempts to distort its shape.
It is by the possession of such a power that solid bodies are
distinguished from fluids, which offer no resistance to distortion;
the idea of Young and Fresnel may therefore be expressed by
the simple statement that the aether behaves as an elastic solid.
After the death of Fresnel this conception was developed in a
brilliant series of memoirs to which our attention must now be
directed.
The elastic-solid theory meets with one obvious difficulty at
the outset. If the aether has the qualities of a solid, how is it that
the planets in their orbital motions are able to journey through
it at immense speeds without encountering any perceptible
resistance ? This objection was first satisfactorily answered by
Sir George Gabriel Stokes* (b. 1819, d. 1903), who remarked
that such substances as pitch and shoemaker's wax, though so
rigid as to be capable of elastic vibration, are yet sufficiently
plastic to permit other bodies to pass slowly through them.
The aether, he suggested, may have this combination of qualities
in an extreme degree, behaving like an elastic solid for vibrations
so rapid as those of light, but yielding like a fluid to the much
slower progressive motions of the planets.
Stokes's explanation harmonizes in a curious way with
Fresnel's hypothesis that the velocity of longitudinal waves in
* Trans. Camb. Phil. Soc., viii, p. 287 (1845).
138 The Aether as an Elastic Solid.
the aether is indefinitely great compared with that of the
transverse waves ; for it is found by experiment with actual
substances that the ratio of the velocity of propagation of
longitudinal waves to that of transverse waves increases
rapidly as the medium becomes softer and more plastic.
In attempting to set forth a parallel between light and the
vibrations of an elastic substance, the investigator is compelled
more than once to make a choice between alternatives. He
may, for instance, suppose that the vibrations of the aether are
executed either parallel to the plane of polarization of the light
or at right angles to it ; and he may suppose that the different
refractive powers of different media are due either to differences
in the inertia of the aether within the media, or to differences
in its power of resisting distortion, or to both these causes
combined. There are, moreover, several distinct methods for
avoiding the difficulties caused by the presence of longitudinal
vibrations ; and as, alas ! we shall see, a further source of
diversity is to be found in that liability to error from which no
man is free. It is therefore not surprising that the list of
elastic-solid theories is a long one.
At the time when the transversality of light was dis-
covered, no general method had been developed for investi-
gating mathematically the properties of elastic bodies; but
under the stimulus of Fresnel's discoveries, some of the best
intellects of the age were attracted to the subject. The volume
of Memoirs of the Academy which contains Fresnel's theory of
crystal-optics contains also a memoir by Claud Louis Marie
Henri Navier* (&. 1785, d. 1836), at that time Professor of
Mechanics in Paris, in which the correct equations of vibratory
motion for a particular type of elastic solid were for the first
time given. ISTavier supposed the medium to be ultimately
constituted of an immense number of particles, which act on
each other with forces directed along the lines joining them, and
depending on their distances apart ; and showed that if e denote
* Mem. de 1'Acad. vii, p. 375. The memoir was presented in 1821, and
published in 1827.
The Aether as an Elastic Solid. 139
the (vector) displacement of the particle whose undisturbed
position is (x, y, z], and if p denote the density of the medium,
the equation of motion is
p — = - 3n grad div e - n curl curl e,
ot
where n denotes a constant which measures the rigidity, or
power of resisting distortion, of the medium. All such elastic
properties of the body as the velocity of propagation of waves
in it must evidently depend on the ratio n/p.
Among the referees of one of Navier's papers was Augustine
Louis Cauchy (b. 1789, d. 1857), one of the greatest analysts of
the nineteenth century,* who, becoming interested in the
question, published in 1828f a discussion of it from an entirely
different point of view. Instead of assuming, as Navier had
done, that the medium is an aggregate of point-centres of force,
and thus involving himself in doubtful molecular hypotheses,
he devised a method of directly studying the elastic properties
of matter in bulk, and by its means showed that the vibrations
of an isotropic solid are determined by the equation
82e (1 4 \ •
p — = - [fc + -n\ grad div e - n curl curl e ;
here n denotes, as before, the constant of rigidity; and the
constant &, which is called the modulus of compression^. denotes
the ratio of a pressure to the cubical compression produced by
it. Cauchy's equation evidently differs from Navier's in that
* Hamilton's opinion, written in 1833, is worth repeating : " The principal
theories of algebraical analysis (under which I include Calculi) require to he
entirely remodelled ; and Cauchy has done much already for this great object.
Poisson also has done much ; but he does not seem to me to have nearly so logical a
mind as Cauchy, great as his talents and clearness are ; and both are in my
judgment very far inferior to Fourier, whom I place at the head of the French
School of Mathematical Philosophy, even above Lagrange and Laplace, though I
rank their talents above those of Cauchy and Poisson." (Life of Sir W. It.
Hamilton, ii, p. 58.)
t Cauchy, Exercices de Mathematiques iii, p. 160 (1828).
J This notation was introduced at a later period, but is used here in order to
avoid subsequent changes.
140 The Aether as an Elastic Solid.
two constants, k and n, appear instead of one. The reason for
this is that a body constituted from point-centres of force in
Navier's fashion has its moduli of rigidity and compression
connected by the relation*
Actual bodies do not necessarily obey this condition; e.g.
for india-rubber, k is much larger than - n ;f and there seems to
o
be no reason why we should impose it on the aether.
In the same year PoissonJ succeeded in solving the diffe-
rential equation which had thus been shown to determine the
wave-motions possible in an elastic solid. The solution, which
is both simple and elegant, may be derived as follows : — Let the
displacement vector e be resolved into two components, of
which one c is circuital, or satisfies the condition
div c = 0,
while the other b is irrotational, or satisfies the condition
curl b = 0.
The equation takes the form
+ 5 Vb = °'
o Tlj
* In order to construct a body whose elastic properties are not limited by this
equation, William John Macquorn Rankine (b. 1820, d. 1872) considered a con-
tinuous fluid in which a number of point-centres of force are situated : the fluid is
supposed to be partially condensed round these centres, the elastic atmosphere of
each nucleus being retained round it by attraction. An additional volume-elasticity
due to the fluid is thus acquired ; and no relation between k and n is now necessary.
Cf. Rankine's Miscellaneous Scientific Papers, pp. 81 sqq.
Sir "William Thomson (Lord Kelvin), in 1889, formed a solid not obeying
Navier's condition by using pairs of dissimilar atoms. Cf. Thomson's Papers,
iii, p. 395. Cf. also Baltimore Lectures, pp. 123 sqq.
t It may, however, be objected that india-rubber and other bodies which
fail to fulfil Navier's relation are not true solids. On this historic controversy,
cf. Todhunter and Pearson's History of Elasticity, i, p. 496.
J Mem. de 1'Acad., viii (1828), p. 623. Poisson takes the equation in the
restricted form given by Navier ; but this does not affect the question of wave-
propagation.
The Aether as an Elastic Solid. 141
The terms which involve b and those which involve c must
be separately zero, since they represent respectively the irrota-
tional and the circuital parts of the equation. Thus, c satisfies.
the pair of equations
02-
p T-J- = ?iV2c, div c = 0 ;
vt
while b is to be determined from
dt
A particular solution of the equations for c is easily seen to be
cx = A sin A (2 - t /-),
/-), cy = B sinXfz - t /-), cz = 0,
\PJ V \PJ
which represents a transverse plane wave propagated with
velocity ^/(n/p). It can be shown that the general solution of
the differential equations for c is formed of such waves as this,
travelling in all directions, superposed on each other
A particular solution of the equations for b is
-t E
V
p
which represents a longitudinal wave propagated with velocity
the general solution of the differential equation for b is formed
by the superposition of such waves as this, travelling in all
directions.
Poisson thus discovered that the waves in an elastic solid
are of two kinds : those in c are transverse, and are propagated
with velocity (n/p)b ; while those in b are longitudinal, and are
propagated with velocity {(k+ $n)/p}%. The latter are* waves
of dilatation and condensation, like sound-waves ; in the c-waves,
on the other hand, the medium is not dilated or condensed, but
* Cf . Stokes, "On the Dynamical Problem of Diffraction," Camb. Phil.
Trans., ix (1849).
142 The Aether as an Elastic Solid.
only distorted in a manner consistent with the preservation of a
constant density.*
The researches which have been mentioned hitherto have
all been concerned with isotropic bodies. Cauchy in 1828f
extended the equations to the case of crystalline substances.
This, however, he accomplished only by reverting to Navier's
plan of conceiving an elastic body as a cluster of particles which
attract each other with forces depending on their distances apart ;
the aelotropy he accounted for by supposing the particles to be
packed more closely in some directions than in others.
The general equations thus obtained for the vibrations of an
elastic solid contain twenty-one constants ; six of these depend
on the initial stress, so that if the body is initially without
stress, only fifteen constants are involved. If, retaining the
initial stress, the medium is supposed to be symmetrical with
respect to three mutually orthogonal planes, the twenty-one
constants reduce to nine, and the equations which determine
the vibrations may be written in the form*
dx\ 2x ty 9 dz
and two similar equations. The three constants G, H, I re-
present the stresses across planes parallel to the coordinate
planes in the undisturbed state of the aether. §
* It may easily be shown that any disturbance, in either isotropic or crystalline
media, for which the direction of vibration of the molecules lies in the wave-front
or surface of constant phase, must satisfy the equation
div 6 = 0,
where e denotes the displacement ; if, on the other hand, the direction of vibration
of the molecules is perpendicular to the wave -front, the disturbance must satisfy
the equation
curl e = 0.
These results were proved by M. O'Brien, Trans. Camb. Phil. Soc., 1842.
t Exercices de Math., iii (1828), p. 188.
J These are substantially equations (68) on page 208 of the third volume of
the Exercices.
$ G, H, I are tensions when they are positive, and pressures when they are
negative.
The Aether as an Elastic Solid. 1 43
On the basis of these equations, Cauchy worked out a
theory of light, of which an instalment relating to crystal-optics
was presented to the Academy in 1830.* Its characteristic
features will now be sketched.
By substitution in the equations last given, it is found that
when the wave-front of the vibration is parallel to the plane
of yz, the velocity of propagation must be (h + G)% if the vibration
takes place parallel to the axis of y, and (g+ G)$ if it takes place
parallel to the axis of z. Similarly when the wave-front is
parallel to the plane of zx, the velocity must be (h + H)% if the
vibration is parallel to the axis of x, and (/+ H)^ if it is parallel
fo the axis of z\ and when the wave-front is parallel to the
plane of xy, the velocity must be (g + /)* if the vibration is parallel
to the axis of x, and (/ + /)* if it is parallel to the axis of y.
Now it is known from experiment that the velocity of a
ray polarized parallel to one of the planes in question is the
same, whether its direction of propagation is along one or the
other of the axes in that plane: so, if we assume that the
vibrations which constitute light are executed parallel to the
plane of polarization, we must have
/+#=/+/, ff + I = g+G, k + H=h+G;
or, G = H=L
This is the assumption made in the memoir of 1830 : the theory
based on it is generally known as Cauchy' s First Theory ;•(• the
equilibrium pressures G, H, /, being all equal, are taken to be zero.
Tf, on the other hand, we make the alternative assumption
that the vibrations of the aether are executed at right angles to
the plane of polarization, we must have
* Mem. de 1'Acad., x, p. 293.
In the previous year (Mem. de 1'Acad., ix, p. 114) Cauchy had stated that the
equations of elasticity lead in the case of uniaxal crystals to a wave-surface of
which two sheets are a sphere and spheroid as in Huygens' theory.
f The equations and results of Cauchy's First Theory of crystal-optics were
independently obtained shortly afterwards hy Franz Ernst Neumann (b. 1798,
d. 1895) : cf. Ann. d. Phys. xxv (1832), p. 418, reprinted as No. 76 of Ostwald's
Klassiker der exakten Wissenschaften, with notes by A. Wangerin.
144 The Aether as an Elastic Solid.
the theory based on this supposition is known as Caucliy's
Second Theory : it was published in 1836.*
In both theories, Cauchy imposes the condition that the
section of two of the sheets of the wave-surface made by any
one of the coordinate planes is to be formed of a circle and an
ellipse, as in Fresnel's theory ; this yields the three conditions
3£c = f(b + c +/) ; 3ca = g(c + a + g) ; Sab = h(a + b + Ji).
Thus in the first theory we have these together with the
equations
£ = 0, H=Q, 1=0,
which express the condition that the undisturbed state of the
aether is unstressed ; and the aethereal vibrations are executed
parallel to the plane of polarization. In the second theory we
have the three first equations, together with
f-Q-h-I-g-H;
and the plane of polarization is interpreted to be the plane at
right angles to the direction of vibration of the aether.
Either of Cauchy's theories accounts tolerably well for the
phenomena of crystal-optics; but the wave-surface (or rather
the two sheets of it which correspond to nearly transverse
waves) is not exactly Fresnel's. In both theories the existence
of a third wave, formed of nearly longitudinal vibrations, is a
formidable difficulty. Cauchy himself anticipated that the
existence of these vibrations would ultimately be demonstrated
by experiment, and in one placef conjectured that they might
be of a calorific nature. A further objection to Cauchy's
theories is that the relations between the constants do not
appear to admit of any simple physical interpretation, being
evidently assumed for the sole purpose of forcing the formulae
into some degree of conformity with the results of experiment.
And further difficulties will appear when we proceed subse-
quently to compare the properties which are assigned to the
aether in crystal- op tics with those which must be postulated in
order to account for reflexion and refraction.
* Comptes Rendus, ii (1836), p. 341 : Mem. de 1'Acad. xviii (1839), p. 153.
f Mem. de 1'Acad. xviii, p. 161.
The Aether as an Elastic Solid. 145
To the latter problem Cauchy soon addressed himself, his
investigations being in fact published* in the same year (1830)
as the first of his theories of crystal-optics.
At the outset of any work on refraction, it is necessary to
assign a cause for the existence of refractive indices, i.e. for the
variation in the velocity of light from one body to another.
Huygens, as we have seen, suggested that transparent bodies
consist of hard particles which interact with the aethereal matter,
modifying its elasticity- Cauchy in his earlier papersf followed
this lead more or less closely, assuming that the density p of the
aether is the same in all media, but that its rigidity n varies
from one medium to another.
Let the axis of x be taken at right angles to the surface of
separation of the media, and the axis of z parallel to the inter-
section of this interface with the incident wave-front; and
suppose, first, that the incident vibration is executed at right
angles to the plane of incidence, so that it may be represented
.by
e~ = /( - x cos i -y sin i + rL t \
where i denotes the angle of incidence ; the reflected wave may
be represented by
ez = FX cos i - y sin i + t
V \/
and the refracted wave by
ez = fi I — x cos r — y sin r + KLt\
where r denotes the angle of refraction, and n' the rigidity of
the second medium.
To obtain the conditions satisfied at the reflecting surface,
Cauchy assumed (without assigning reasons) that the x- and
^/-components of the stress across the #y-plane are equal in
* Bull, des Sciences Math. xiv. (1830), p. 6.
t As will appear, his views on this subject subsequently changed.
L
146 The Aether as an Elastic Solid.
the media on either side the interface. This implies in the
present case that the quantities
tie* dez
n — and n —
dx ty
are to be continuous across the interface : so we have
n cos i'. (/' - 1") = n' cos r . /', ; n sin i.(f' + F) = n' sin r . f\.
Eliminating /'„ we have
F' _ sin (r - i)
f sin (r + i)
Now this is Fresnel's sine-law for the ratio of the intensity
of the reflected ray to that of the incident ray ; and it is known
that the light to which it applies is that which is polarized
parallel to the plane of incidence. Thus Cauchy was driven
to the conclusion that, in order to satisfy the known facts
of reflexion and refraction, the vibrations of the aether must be
supposed executed at right angles to the plane of polarization
of the light.
The case of a vibration performed in the plane of incidence
he discussed in the same way. It was found that Fresnel's
tangent-law could be obtained by assuming that ex and the
normal pressure across the interface have equal values in the
two contiguous media.
The theory thus advanced was encumbered with many diffi-
culties. In the first place, the identification of the plane of
polarization with the plane at right angles to the direction of
vibration was contrary to the only theory of crystal-optics which
Cauchy had as yet published. In the second place, no reasons
were given for the choice of the conditions at the interface.
Cauchy's motive in selecting these particular conditions was
evidently to secure the fulfilment of Fresnel's sine-law and
tangent-law; but the results are inconsistent with the true
boundary-conditions, which were given later by Green.
It is probable that the results of the theory of reflexion had
much to do with the decision, which Cauchy now made,* to
*Comptes Rendus, ii. (1836), p. 341.
The Aether as an Elastic Solid. 147
reject the first theory of crystal-optics in favour of the second.
After 1836 he consistently adhered to the view that the vibra-
tions of the aether are performed at right angles to the plane of
polarization. In that year he made another attempt to frame a
satisfactory theory of reflexion,* based on the assumption just
mentioned, and on the following boundary-conditions: — At
the interface between two media curl e is to be continuous, and
(taking the axis of x normal to the interface) dex/dx is also to
be continuous.
Again we find no very satisfactory reasons assigned for the
choice of the boundary- conditions ; and_as the continuity of e
itself across the interface is not included amongst the conditions
cHosen, they are obviously open to criticism ; but they lead to
Fresnel's sine- and tangent-equations, which correctly express
the actual behaviour of light. f Cauchy remarks that in order to
justify them it is necessary to abandon the assumption of his
earlier theory, that the density of the aether is the same in all
material bodies.
It may be remarked that neither in this nor in Cauchy's
earlier theory of reflexion is any trouble caused by the appear-
ance of longitudinal waves when a transverse wave is reflected,
for the simple reason that he assumes the boundary-conditions to
be only four in number ; and these can all be satisfied without
the necessity for introducing any but transverse vibrations.
These features bring out the weakness of Cauchy's method of
attacking the problem. His object was to derive the properties
of light from a theory of the vibrations of elastic solids. At the
outset he had already in his possession the differential equations
of motion of the solid, which were to be his starting-point, and
the equations of Fresnel, which were to be his goal. It only
* Comptes Rendus, ii. (1836), p. 341 : " Meraoire sur la dispersion delalumiere "
(Nouveaux exercices de Math., 1836), p. 203.
t These boundary -conditions of Cauchy's are, as a matter of fact, satisfied by
the electric force in the electro-magnetic theory of light. The continuity of
<;url e is equivalent to the continuity of the magnetic vector across the interface,
and the continuity of (tex/dx leads to the same equation as the continuity of
the component of electric force in the direction of the intersection of the
interface with the plane of incidence.
L 2
] 48 The Aether as an Elastic Solid.
remained to supply the boundary-conditions at an interface,
which are required in the discussion of reflexion, and the
relations between the elastic constants of the solid, which are
required in the optics of crystals. Cauchy seems to have con-
sidered the question from the purely analytical point of view.
Given certain differential equations, what supplementary con-
ditions must be adjoined to them in order to produce a given
analytical result ? The problem when stated in this form
admits of more than one solution ; and hence it is not surprising
that within the space of ten years the great French mathe-
matician produced two distinct theories of crystal-optics and
three distinct theories of reflexion,* almost all yielding correct
or nearly correct final formulae, and yet mostly irreconcilable
with each other, and involving incorrect boundary-conditions
and improbable relations between elastic constants.
Cauchy's theories, then, resemble Fresnel's in postulating
types of elastic solid which do not exist, and for whose
assumed properties no dynamical justification is offered. The
same objection applies, though in a less degree, to the original
form of a theory of reflexion and refraction which was,
discovered about this timef almost simultaneously by James
MacCullagh (6. 1809, d. 1847), of Trinity College, Dublin,
and Franz Neumann (b. 1798, d. 1895), of Konigsberg. To
these authors is due the merit of having extended the laws
of reflexion to crystalline media; but the principles of the
theory were originally derived in connexion with the simpler
ease of isotropic media, to which our attention will for the
present be confined.
* One yet remains to be mentioned.
f The outlines of the theory were published by MacCullagh in Brit. Assoc. Rep.
1835 ; and his results were given in Phil. Mag. x (Jan., 1837), and in Proc.
Royal Irish Acad. xviii. (Jan., 1837). Neumann's memoir was presented to the
Berlin Academy towards the end of 1835, and published in 1837 in Abh. Berl.
Ak. aus dem Jahre 1835, Math. Klasse, p. 1. So far as publication is concerned,
the priority would seem to belong to MacCullagh; but there are reasons for
believing that the priority of discovery really rests with Neumann, who had
arrived at his equations a year before they were communicated to the Berlin
Academy.
The Aether as an Elastic Solid. 149
MacCullagh and Neumann felt that the great objection
to FresnePs theory of reflexion was its failure to provide for
the continuity of the normal component of displacement at the
interface between two media ; it is obvious that a discontinuity
in this component could not exist in any true elastic-solid
theory, since it would imply that the two media do not remain
in contact. Accordingly, they made it a fundamental con-
dition that all three components of the displacement must be
continuous at the interface, and found that the sine-law and
tangent-law can be reconciled with this condition only by
supposing that the aether- vibrations are parallel to the plane of
polarization : which supposition they accordingly adopted. In
place of the remaining three true boundary-conditions, however,
they used only a single equation, derived by assuming that
transverse incident waves give rise only to transverse reflected
and refracted waves, and that the conservation of energy holds
for these — i.e. that the masses of aether put in motion,
multiplied by the squares of the amplitudes of vibration, are
the same before and after incidence. This is, of course, the
same device as had been used previously by Presnel; it
must, however, be remarked that the principle is unsound as
applied to an ordinary elastic solid; for in such a body the
refracted and reflected energy would in part be carried away
by longitudinal waves.
In order to obtain the sine and tangent laws, MacCullagh
and Neumann found it necessary to assume that the inertia
of the luminiferous medium is everywhere the same, and
that the differences in behaviour of this medium in different
substances are due to differences in its elasticity. The two
laws may then be deduced in much the same way as in the
previous investigations of Fresnel and Cauchy.
Although to insist on continuity of displacement at the
interface was a decided advance, the theory of MacCullagh and
Neumann scarcely showed as yet much superiority over the
quasi-mechanical theories of their predecessors. Indeed,
MacCullagh himself expressly disavowed any claim to regard
150 The Aether as an Elastic Solid.
his theory, in the form to which it had then been brought, as a
final explanation of the properties of light. " If we are asked,"
he wrote, " what reasons can be assigned for the hypotheses on
which the preceding theory is founded, we are far from being
able to give a satisfactory answer. We are obliged to confess
that, with the exception of the law of vis viva, the hypotheses
are nothing more than fortunate conjectures. These conjectures
are very probably right, since they have led to elegant laws
which are fully borne out by experiments ; but this is all we
can assert respecting them. We cannot attempt to deduce
them from first principles ; because, in the theory of light,
such principles are still to be sought for. It is certain, indeed,
that light is produced by undulations, propagated, with
transversal vibrations, through a highly elastic aether ; but the
constitution of this aether, and the laws of its connexion (if it
has any connexion) with the particles of bodies, are utterly
unknown/'
The needful reformation of the elastic-solid theory of
reflexion was effected by Green, in a paper* read to the
Cambridge Philosophical Society in December, 1837. Green,
though inferior to Cauchy as an analyst, was his superior in
physical insight ; instead of designing boundary-equations for
the express purpose of yielding Fresnel's sine and tangent
formulae, he set to work to determine the conditions which are
actually satisfied at the interfaces of real elastic solids.
These he obtained by means of general dynamical principles.
In an isotropic medium which is strained, the potential energy
per unit volume due to the state of stress is
4 \tex dey
+ (~- + ^} -4r-*~-4~~-4
where e denotes the displacement, and k and n denote the two
* Trans. Camb. Phil. Soc., 1838 ; Green's Math. Papers, p. 245.
The Aether as an, Elastic Solid. 151
elastic constants already introduced; by substituting this value
of <f> in the general variational equation
III'0 \w &t + 1* ** + TF*-| ****** = -
* (where p denotes the density), the equation of motion may be
deduced.
But this method does more than merely furnish the equation
of motion
or,
/ 4 \
pe = - ( k + - n ) grad div e - n curl curl e ;
\ • /
pe = -lk + -n\ grad div e + nVze,
which had already been obtained by Cauchy ; for it also yields
the boundary-conditions which must be satisfied at the interface
between two elastic media in contact ; these are, as might be
guessed by physical intuition, that the three components of the
displacement* and the three components of stress across the
interface are to be equal in the two media. If the axis of x
be taken normal to the interface, the latter three quantities
are
, 2 \ dex fiez 3ex\ fdev dey
--TI dive+ 27i — , w(-Ji + — ), and n (^ + -£
3 ) dx \to fa J \ty dx
The correct boundary-conditions being thus obtained, it was
a simple matter to discuss the reflexion and refraction of an
incident wave by the procedure of Fresnel and Cauchy. The
result found by Green was that if the vibration of the aethereal
molecules is executed at right angles to the plane of incidence,
the intensity of the reflected light obeys Fresnel's sine-law, pro-
vided the rigidity n is assumed to be the same for all media,
but the inertia p to vary from one medium to another. Since
the sine-law is known to be true for light polarized in the plane
of incidence, Green's conclusion confirmed the hypotheses of
* These first three conditions are of course not dynamical but geometrical.
152 The Aether as an Elastic Solid.
Fresnel, that the vibrations are executed at right angles to the
plane of polarization, and that the optical differences between
media are due to the different densities of aether within them.
It now remained for Green to discuss the case in which the
incident light is polarized at right angles to the plane of inci-
dence, so that the motion of the aethereal particles is parallel to
the intersection of the plane of incidence with the front of the
wave. In this case it is impossible to satisfy all the six
boundary-conditions without assuming that longitudinal vibra-
tions are generated by the act of reflexion. Taking the plane
of incidence to be the plane of yz, and the interface to be the
plane of xy, the incident wave may be represented by the
equations
6 = A + lz
where, if i denote the angle of incidence, we have
I = . /— cos it m = - /— sin i.
\n Mn
There will be a transverse reflected wave,
and a transverse refracted wave,
y); ez = - C — f(t + 1& + my),
where, since the velocity of transverse waves in the second
medium is v/W/oz, we can determine ^ from the equation
^•f^.&j
n
there will also be a longitudinal reflected wave,
8 9
ey = D -f(t -\z + my); ez = D -f(t - \z + my),
The Aeiher as an Elastic Solid. 153
where A is determined by the equation
and a longitudinal refracted wave,
7\ 7\
ey = JE - /(* + Aiz + my) ; ez = E - f(t
where AI is determined by
Substituting these values for the displacement in the boundary-
conditions which have been already formulated, we obtain the
equations which determine the intensities of the reflected and
refracted waves ; in particular, it appears that the amplitude of
the reflected transverse wave is given by the equation
A- E _ ljj>i m? (pi - p2)2
A + B Ip2 I pz (\pz + A!/?I)
Now if the elastic constants of the media are such that the
velocities of propagation of the longitudinal waves are of the
same order of magnitude as those of the transverse waves, the
direction-cosines of the longitudinal reflected and refracted rays
will in general have real values, and these rays will carry away
some of the energy which is brought to the interface by the
incident wavev-G^een avoided this difficulty by adopting Fresnel's
suggestion that the resistance of the aether to compression may V\
be very large in comparison with the resistance to distortion, \\
as is actually the case with such substances as jelly and
caoutchouc : in this case the longitudinal waves are degraded in
much the same way as the transverse refracted ray is degraded
when there is total reflexion, and so do not carry away energy.
Making this supposition, so that k\ and &2 are very large, the
•quantities A and A: have the values m </ - 1, and we have
A- B li pi m (PI - p2f
A + B I 2
154 The Aether as an Elastic Solid.
Thus if BjA denote the modulus of £/A, we have
p\
if>l
This expression represents the ratio of the intensity of the
transverse reflected wave to that of the incident wave. It
does not agree with Fresnel's tangent- formula : and both on this
account and also because (as we shall see) this theory of reflexion
does not harmonize well with the elastic-solid theory of crystal-
optics, it must be concluded that the vibrations of a Greenian
solid do not furnish an exact parallel to the vibrations which
constitute light.
The success of Green's investigation from the standpoint of
dynamics, set off by its failure in the details last mentioned,
stimulated MacCullagh to fresh exertions. At length he succeeded
in placing his own theory, which had all along been free from
reproach so far as agreement with optical experiments was
concerned, on a sound dynamical basis ; thereby effecting that
reconciliation of the theories of Light and Dynamics which had
been the dream of every physicist since the days of Descartes.
The central feature of MacCullagh's investigation,* which
was presented to the Eoyal Irish Academy in 1839, is the intro-
duction of a new type of elastic solid. He had, in fact, concluded
from Green's results that it was impossible to explain optical
phenomena satisfactorily by comparing the aether to an elastic
solid of the ordinary type, which resists compression and
distortion ; and he saw that the only hope of the situation was
to devise a medium which should be as strictly conformable to-
dynamical laws as Green's elastic solid, and yet should have
its properties specially designed to fulfil the requirements of
the theory of light. Such a medium he now described.
If as before we denote by e the vector displacement of a
point of the medium from its equilibrium position, it is well
* Trans. Roy. Irish Acad. xxi. : MacCullagh's Coll. Works, p. 145.
The Aether as an Elastic Solid. 155
known that the vector curl e denotes twice the rotation of the
part of the solid in the neighbourhood of the point (x, y, z) from
its equilibrium orientation. In an ordinary elastic solid, the
potential energy of strain depends only on the change of size
and shape of the volume- elements ; on their compression and
distortion, in fact. For MacCullagh's new medium, on the
other hand, the potential energy depends only on the rotation
of the volume-elements.
Since the medium is not supposed to be in a state of stress
in its undisturbed condition, the potential energy per unit
volume must be a quadratic function of the derivates of e ; so
that in an isotropic medium this quantity <f> must be formed
from the only invariant which depends solely on the rotation
and is quadratic in the derivates, that is from (curl e)2 ; thus
we may write
to,
*~
The equation of motion is now to be determined, as in the
case of Green's aether, from the variational equation
the result is
p—z = - fi curl curl e.
It is evident from this equation that if div e is initially
zero it will always be zero: we shall suppose this to be the
case, so that no longitudinal waves exist at any time in the
medium. One of the greatest difficulties which beset elastic-
solid theories is thus completely removed.
The equation of motion may now be written
156 The Aether as an Elastic Solid.
which shows that transverse waves are propagated with velocity
From the variational equation we may also determine the
boundary-conditions which must be satisfied at the interface
between two media ; these are, that the three components of e
are to be continuous across the interface, and that the two
components of p curl e parallel to the interface are also to be
continuous across it. One of these five conditions, namely, the
continuity of the normal component of e, is really dependent on
the other four ; for if we take the axis of x normal to the
interface, the equation of motion gives
p a"? = ~3^ (» curl e)* + l°* curl e)" / ' . |
and as the quantities p, (n curl e)2, and (/n curl e)y are continuous
across the interface, the continuity of c>2ex/dtz follows. Thus the
only independent boundary-conditions in MacCullagh's theory
are the continuity of the tangential components of e and of
fj curl e.* It is easily seen that these are equivalent to the
boundary-conditions used in MacCullagh's earlier paper, namely,
the equation of vis viva and the continuity of the three
components of e : and thus the " rotationally elastic " aether of
this memoir furnishes a dynamical foundation for the memoir
of 1837.
The extension to crystalline media is made by assuming the
potential energy per unit volume to have, when referred to the
principal axes, the form
\dz dx J \c>x ty J
where A, B, C denote three constants which determine the
optical behaviour of the medium : it is readily seen that the
wave-surface is Fresnel's, and that the plane of polarization
* MacCullagh's equations may readily be interpreted in the electro -magnetic
theory of light : e corresponds to the magnetic force, p curl e to the electric force,
and curl e to the electric displacement.
The Aether as an Elastic Solid. 157
contains the displacement, and is at right angles to the
rotation.
MacCullagh's work was regarded with doubt by his own
and the succeeding generation of mathematical physicists, and
can scarcely be said to have been properly appreciated until
FitzGerald drew attention to it forty years afterwards. But
there can be no doubt that MacCullagh really solved the
problem of devising a medium whose vibrations, calculated in
accordance with the correct laws of dynamics, should have the
same properties as the vibrations of light.
The hesitation which was felt in accepting the rotationally
elastic aether arose mainly from the want of any readily
conceived example of a body endowed with such a property.
This difficulty was removed in 1889 by Sir William Thomson
(Lord Kelvin), who designed mechanical models possessed of
rotational elasticity. Suppose, for example,* that a structure is
formed of spheres, each sphere being in the centre of the
tetrahedron formed by its four nearest neighbours. Let each
sphere be joined to these four neighbours by rigid bars, which
have spherical caps at their ends so as to slide freely on the
spheres. Such a structure would, for small deformations, behave
like an incompressible perfect fluid. Now attach to each bar a
pair of gyroscopically-mounted flywheels, rotating with equal
and opposite angular velocities, and having their axes in the line
of the bar : a bar thus equipped will require a couple to hold
it at rest in any position inclined to its original position, and
the structure as a whole will possess that kind of quasi-
elasticity which was first imagined by MacCullagh.
This particular representation is not perfect, since a system
of forces would be required to hold the model in equilibrium if
it were irrotationally distorted. Lord Kelvin subsequently
invented another structure free from this defect. t
* Comptes Eendus, Sept. 16, 1889 : Kelvin's Math, and Phys. Papers, iii,
p. 466.
tProc. Roy. Soc. Edinb., Mar. 17, 1890: Kelvin's Math, and Phys. Papers,
iii, p. 468.
158 The Aether as an Elastic Solid.
The work of Green proved a stimulus not only to
MacCullagh but to Cauchy, who now (1839) published yet
a third theory of reflexion.* This appears to have owed its
origin to a remark of Green's, f that the longitudinal wave
might be avoided in either of two ways — namely, by supposing
its velocity to be indefinitely great or indefinitely small. Green
curtly dismissed the latter alternative and adopted the former,
on the ground that the equilibrium of the medium would be
unstable if its compressibility were negative (as it must be if
the velocity of longitudinal waves is to vanish). Cauchy, without
attempting to meet Green's objection, took up the study of a
medium whose elastic constants are connected by the equation
k + ±n = 0,
so that the longitudinal vibrations have zero velocity; and showed
that if the aethereal vibrations are supposed to be executed at
right angles to the plane of polarization, and if the rigidity
•of the aether is assumed to be the same in all media, a ray
which is reflected will obey the sine-law and tangent-law of
Fresnel. The boundary-conditions which he adopted in order to
obtain this result were the continuity of the displacement e and
-of its derivate 3e/9#, where the axis of x is taken at right
angles to the interface.* These are not the true boundary-con-
ditions for general elastic solids ; but in the particular case now
under discussion, where the rigidity is the same in the two
media, they yield the same equations as the conditions correctly
given by Green.
The aether of Cauchy's third theory of reflexion is well
worthy of some further study. It is generally known as they
.contractile or labile^ aether, the names being due to William
* Comptes Rendus, ix, p. 676 (25 NOT., 1839), and p. 726 (2 Dec., 1839).
t Green's Math. Papers, p. 246.
J Comptes Eendus, x, p. 347 (March 2, 1840) : xxvii, p. 621 (1848) ; sxviii, p. 25
(1849). Mem. de 1'Acad., xxii (1848), pp. 17, 29.
§ Labile or neutral is a term used of such equilibrium as that of a rigid body oil
-a perfectly smooth horizontal plane.
The Aether as an Elastic Solid. 159
Thomson (Lord Kelvin), who discussed it long afterwards.* It
may be defined as an elastic medium of (negative) com-
pressibility such as to make the velocity of the longitudinal
wave zero : this implies that no work is required to be done
in order to give the medium any small irrotational disturbance.
An example is furnished by homogeneous foam free from air
and held from collapse by adhesion to a containing vessel
Cauchy, as we have seen, did not attempt to refute Green's
objection that such a medium would be unstable ; but, as
Thomson remarked, every possible infinitesimal motion of the
medium is, in the elementary dynamics of the subject, proved
to be resolvable into coexistent wave-motions. If, then, the
velocity of propagation for each of the two kinds of wave-motion
is real, the equilibrium must be stable, provided the medium
either extends through boundless space or has a fixed containing
vessel as its boundary.
When the rigidity of the luminiferous medium is supposed
to have the same value in all bodies, the conditions to be satisfied
at an interface reduce to the continuity of the displacement e,
of the tangential components of curl e, and of the scalar
quantity (k + ^n) div e across the interface.
Now we have seen that when a transverse wave is incident
on an interface, it gives rise in general to reflected and refracted
waves of both the transverse ajid the longitudinal species. In
the case of the contractile aether, for which the velocity of
propagation of the longitudinal waves is very small, the ordinary
construction for refracted waves shows that the directions of
propagation of the reflected and refracted longitudinal waves
will be almost normal to the interface. The longitudinal
waves will therefore contribute only to the component of
displacement normal to the interface, not to the tangential
components : in other words, the only tangential components of
displacement at the interface are those due to the three trans-
verse waves — the incident, reflected, and refracted. Moreover,
the longitudinal waves do not contribute at all to curl e ; and,
* Phil. Mag. xxvi (1888), p. 414.
160 The Aether as an Elastic Solid.
therefore, in the contractile aether, the conditions that the
tangential components of e and of n curl e shall be continuous
across an interface are satisfied by the distortional part of the
disturbance taken alone. The condition that the component
of e normal to the interface is to be continuous is not satisfied
by the distortional part of the disturbance taken alone, but is
satisfied when the distortional and congressional parts are taken
together.
The energy carried away by the longitudinal waves is
infinitesimal, as might be expected, since no work is required in
order to generate an irrotational displacement. Hence, with
this aether, the behaviour of the transverse waves at an
interface may be specified without considering the irrotational
part of the disturbance at all, by the conditions that the
conservation of energy is to hold and that the tangential
components of e and of n curl e are to be continuous. But if
we identify these transverse waves with light, assuming that
the displacement e is at right angles to the plane of polarization
of the light, and assuming moreover that the rigidity n is the
same in all media* (the differences between media depending on
differences in the inertia p), we have exactly the assumptions
of Fresnel's theory of light : whence it follows that transverse
waves in the labile aether must obey in reflexion the sine-law
and tangent-law of Fresnel.
The great advantage of the labile aether is that it overcomes
the difficulty about securing continuity of the normal com-
ponent of displacement at an interface between two media :
the light-waves taken alone do not satisfy this condition of
continuity ; but the total disturbance consisting of light- waves
and irrotational disturbance taken together does satisfy it ;
and this is ensured without allowing the irrotational disturbance
to carry off any of the energy. f
* This condition is in any case necessary for stability, as was shown by
R. T. Glazebrook : cf. Thomson, Phil. Mag. xxvi, p. 500.
f The labile-aether theory of light may be compared with the electro-magnetic
theory, by interpreting the displacement e as the electric force, and pe as the
electric displacement.
The Aether as an Elastic Solid. 161
William Thomson (Lord Kelvin, b. 1824, d. 1908), who
devoted much attention to the labile aether, was at one time
led to doubt the validity of this explanation of light* ; for when
investigating the radiation of energy from a vibrating rigid
globe embedded in an infinite elastic-solid aether, he found that
in some cases the irrotational waves would carry away a
considerable part of the energy if the aether were of the labile
type. This difficulty, however, was removed by the observationf
that it is sufficient for the fulfilment of Fresnel's laws if the
velocity of the irrotational waves in one of the two media is
very small, without regard to the other medium. Following up
this idea, Thomson assumed that in space void of ponderable
matter the aether is practically incompressible by the forces
concerned in light-waves, but that in the space occupied by
liquids and solids it has a negatiye_CQmpres^biIiljy , so as to give
zero velocity for longitudinal aether- waves in these bodies.
This assumption was based on the conception that material
atoms move through space without displacing the aether: a
conception which, as Thomson remarked, contradicts the old
scholastic axiom that two different portions of matter cannot
simultaneously occupy the same space.J He supposed the
aether to be attracted and repelled by the atoms, and thereby to
be condensed or rarefied. §
The year 1839, which saw the publication of MacCullagh's
dynamical theory of light and Cauchy's theory of the labile
aether, was memorable also for the appearance of a memoir by
Green on crystal-bptics.H This really contains two distinct
theories, which respectively resemble Cauchy's First and Second
Theories : in one of them, the stresses in the undisturbed state
* Baltimore Lectures (edition 1904), p. 214.
t Ibid. (ed. 1904), p. 411.
^ Michell and Boscovich in the eighteenth century had taught the doctrine of
the mutual penetration of matter, i.e. that two substances may be in the same
place at the same time without excluding each other : cf. Priestley's History i.,
p. 392.
6 Cf. Baltimore Lectures (ed. 1904), pp. 413-14, 463, and Appendices A and E.
|j Cambridge Phil. Trans., 1839 ; Green's Math. Papers;?. 293.
M
162 The Aether as an Elastic Solid.
of the aether are supposed to vanish, and the vibrations of the
aether are supposed to be executed parallel to the plane of
polarization of the light ; in the other theory, the initial stresses
are not supposed to vanish, and the aether- vibrations are at
right angles to the plane of polarization. The two investigations
are generally known as Green's First and Second Theories of
crystal-optics.
The foundations of both theories are, however, the same.
Green first of all determined the potential energy of a strained
crystalline solid ; this in the most general case involves 27
constants, or 21 if there is no initial stress.* If, however, as is
here assumed, the medium possesses three planes of symmetry
at right angles to each other, the number of constants reduces
to. 12, or to 9 if there is no initial stress; if e denote the dis-
placement, the potential energy per unit volume may be written
fo \2 ("be \2 fr\f \i\ (ffo \2 /f)p \2 /a/, N
+ $} * (£) 1 + ** ft) + (I) + (I
fty*.
7 ty tz 9
sf3ey 3g,
•*• 4/1 ;? + s-
2</ \dz ty
*
dx
The usual variational equation
= - [[f
* For there are 21 terms in a homogeneous function of the second degree in six
variables.
The Aether as an Elastic Solid. 163
then yields the differential equations of motion, namely :
8 / dex dey fe,\ a / aex a^ , a^\
+ — a— + h ^ + g — + — a — + A — + 0 — ],
acVSaj ty y dzj dx\ fa ty y fa)9
and two similar equations.
These differ from Cauchy's fundamental equations in having
greater generality: for Cauchy's medium was supposed to be
built up of point-centres of force attracting each other according
to some function of the distance ; and, as we have seen, there
are limitations in this method of construction, which render it
incompetent to represent the most general type of elastic solid.
Cauchy's equations for crystalline media are, in fact, exactly
analogous to the equations originally found by Navier for
isotropic media, which contain only one elastic constant instead
of two.
The number of constants in the above equations still exceeds
the three which are required to specify the properties of a
biaxal crystal : and Green now proceeds to consider how the
number may be reduced. The condition which he imposes for
this purpose is that for two of the three waves whose front is
parallel to a given plane, the vibration of the aethereal molecules
shall be accurately in the plane of the wave : in other words,
that two of the three waves shall be purely distortional, the
remaining one being consequently a normal vibration. This
condition gives five relations,* which may be written : —
a. « b = c = JJK;
/'-j»-2/ / = M-2<7; tf-M-2fc;
where /z denotes a new constant, f
* As Green showed, the hypothesis of transversality really involves the existence
of planes of symmetry, so that it alone is capable of giving 14 relations between the
21 constants : and 3 of the remaining 7 constants may be removed by change of
axes, leaving only four.
t It was afterwards shown by Barre de Saint- Venant (b. 1797, d. 1886),
Journal de Math., vii (1863), p. 399, that if the initial stresses be supposed to
vanish, the conditions which must be satisfied among the remaining nine constants
M 2
164 The Aether as an Elastic Solid.
Thus the potential energy per unit volume may be written
. n^* ^ rrdey^. Tde*
d> = 6r — + ±L — + -I ^~
ox oy oz
i T ] Wx\ ^ ™y\ -L. oe*
I1 Ur + Ur- + Ur
At this point Green's two theories of crystal-optics diverge
from each other. According to the first theory, the initial
stresses G-, H, I are zero, so that
", *> «>/>#>*>/' /, *'» in order that the wave-surface may be Fresnel's, are the
following : —
(34 _/) (3c -/)=(/ + /')*
((3a-
(Za - h) (3£ - h) = (h + A')
These reduce to Green's relations when the additional equation b = c is assumed.
Saint-Venant disputed the validity of Green's relations, asserting that they?are
compatible only with isotropy. On this controversy cf. E. T. Glazebrook, Brit.
Assoc. Report, 1885, p. 171, and Karl Pearson in Todhunter and Pearson's History
of Elasticity, ii, § 147.
The Aether as an Elastic Solid. 165
This expression contains the correct number of constants,
namely, four: three of them represent the optical constants
of a biaxal crystal, and one (namely, ju) represents the square of
the velocity of propagation of longitudinal waves. It is found
that the two sheets of the wave-surface which correspond to the
two distortional waves form a Fresnel's wave-surface, the third
sheet, which corresponds to the longitudinal wave, being an
ellipsoid. The directions of polarization and the wave- velocities
of the distortional waves are identical with those assigned by
Fresnel, provided it is assumed that the direction of vibration
of the aether- particles is parallel to the plane of polarization ;
but this last assumption is of course inconsistent with Green's
theory of reflexion and refraction.
In his Second Theory, Green, like Cauchy, used the condition
that for the waves whose fronts are parallel to the coordinate
planes, the wave- velocity depends only on the plane of polariza-
tion, and not on the direction of propagation. He thus obtained
the equations already found by Cauchy —
O-f-H-g-I-h.
The wave-surface in this case also is Fresnel's, provided it
is assumed that the vibrations of the aether are executed at
right angles to the plane of polarization.
The principle which underlies the Second Theories of Green
and Cauchy is that the aether in a crystal resembles an elastic
solid which is unequally pressed or pulled in different directions
by the unmoved ponderable matter. This idea appealed strongly
to W. Thomson (Kelvin), who long afterwards developed it
further,* arriving at the following interesting result : — Let an
incompressible solid, isotropic when unstrained, be such that its
potential energy per unit volume is
P 7
where q denotes its modulus of rigidity when unstrained, and
* Proc. R. S. Edin. xv (1887), p. 21 : Phil. Mag. xxv (1888) p. 116 : Baltimore
Lectures (ed. 1904), pp. 228-259.
1 66 The Aether as an Elastic Solid.
«*> j3*> 7*> denote the proportions in which lines parallel to the
axes of strain are altered ; then if the solid be initially strained
in a way defined by given values of a, (3, y, by forces applied to
its surface, and if waves of distortion be superposed on this
initial strain, the transmission of these waves will follow exactly
the laws of Fresnel's theory of crystal- optics, the wave-surface
being
*
q q
There is some difficulty in picturing the manner in which
the molecules of ponderable matter act upon the aether so as to
produce the initial strain required by this theory. Lord
Kelvin utilized* the suggestion to which we have already
referred, namely, that the aether may pervade the atoms of
matter so as to occupy space jointly with them, and that its
interaction with them may consist in attractions and repulsions
exercised throughout the regions interior to the atoms. These
forces may be supposed to be so large in comparison with those
called into play in free aether that the resistance to compres-
sion may be overcome, and the aether may be (say) condensed
in the central region of an isolated atom, and rarefied in its
outer parts. A crystal may be supposed to consist of a group
of spherical atoms in which neighbouring spheres overlap each
other ; in the central regions of the spheres the aether will be
condensed, and within the lens-shaped regions of overlapping
it will be still more rarefied than in the outer parts of a solitary
atom, while in the interstices between the atoms its density
will be unaffected. In consequence of these rarefactions and
condensations, the reaction of the aether on the atoms tends
to draw inwards the outermost atoms of the group, which,
however, will be maintained in position by repulsions between
the atoms themselves; and thus we can account for the pull
which, according to the present hypothesis, is exerted on the
aether by the ponderable molecules of crystals.
* Baltimore Lectures (ed. 1904), p. 253.
The Aether as an Elastic Solid. 167
Analysis similar to that of Cauchy's and Green's Second
Theory of crystal-optics may be applied to explain the doubly
refracting property which is possessed by strained glass ; but
in this case the formulae derived are found to conflict with
the results of experiment. The discordance led Kelvin to
doubt the truth of the whole theory. "After earnest and
hopeful consideration of the stress theory of double refraction
during fourteen years," he said,* " I am unable to see how it
can give the true explanation either of the double refraction of
natural crystals, or of double refraction induced in isotropic
solids by the application of unequal pressures in different
directions."
It is impossible to avoid noticing throughout all Kelvin's
work evidences of the deep impression which was made
upon him by the writings of Green. The same may be said
of Kelvin's friend and contemporary Stokes; and, indeed, it
is no exaggeration to describe Green as the real founder of
that " Cambridge school " of natural philosophers, of which
Kelvin, Stokes, Lord Eayleigh, and Clerk Maxwell were the
most illustrious members in the latter half of the nineteenth
century, and which is now led by Sir Joseph Thomson and
Sir Joseph Larrnor. In order to understand the peculiar
position occupied by Green, it is necessary to recall some-
thing of the history of mathematical studies at Cambridge.
The century which elapsed between the death of Newton
and the scientific activity of Green was the darkest in the
history of the University. It is true that Cavendish and
Young were educated at Cambridge; but they, after taking
undergraduate courses, removed to London. In the entire
period the only natural philosopher of distinction who lived
and taught at Cambridge was Michell ; and for some reason
which at this distance of time it is difficult to understand
fully, Michell's researches seem to have attracted little or no
attention among his collegiate contemporaries and successors,
* Baltimore Lectures (ed. 1904), p. 258.
168 The Aether as an Elastic Solid.
who silently acquiesced when his discoveries were attributed to
others, and allowed his name to perish entirely from Cambridge
tradition.
A few years before Green published his first paper, a
notable revival of mathematical learning swept over the
University ; the fluxional symbolism, which since the time of
Newton had isolated Cambridge from the continental schools,
was abandoned in favour of the differential notation, and the
works of the great French analysts were introduced and
eagerly read. Green undoubtedly received his own early
inspiration from this source ; but in clearness of physical
insight and conciseness of exposition he far excelled his
masters ; and the slight volume of his collected papers has
to this day a charm which is wanting to the voluminous
writings of Cauchy and Poisson. It was natural that such an
example should powerfully influence the youthful intellects of
Stokes — who was an undergraduate when Green read his memoir
on double refraction to the Cambridge Philosophical Society—
and of William Thomson (Kelvin), who came into residence two
years afterwards.*
In spite of the advances which were made in the great
memoirs of the year 1839, the fundamental question as to
whether the aether-particles vibrate parallel or at right angles
to the plane of polarization was still unanswered. More light
was thrown on this problem ten years later by Stokes's inves-
tigation of Diffraction.f Stokes showed that on almost any
conceivable hypothesis regarding the aether, a disturbance in
which the vibrations are executed at right angles to the plane
of diffraction must be transmitted round the edge of an opaque
body with less diminution of intensity than a disturbance whose
vibrations are executed parallel to that plane. It follows that
when light, of which the vibrations are oblique to the plane of
*It was in the year Thomson took his degree (1845) that he bought, and read
with delight, the electrical memoir which Green had published at Nottingham in
1828.
f Trans. Camb. Phil. Soc., ix (1849), p. 1. Stokes's Math, and Phys. Papers,
ii, p. 243.
The Aether as an Elastic Solid. 169
diffraction, is so transmitted, the plane of vibration will be more
nearly at right angles to the plane of diffraction in the diffracted
than in the incident light. Stokes himself performed experi-
ments to test the matter, using a grating in order to obtain
strong light diffracted at a large angle, and found that when
the plane of polarization of the incident light was oblique to the
plane of diffraction, the plane of polarization of the diffracted
light was more nearly parallel to the plane of diffraction. This
result, which was afterwards confirmed by L. Lorenz,* appeared
to confirm decisively the hypothesis of Fresnel, that the vibra-
tions of the aethereal particles are executed at right angles to
the plane of polarization.
Three years afterwards Stokes indicatedf a second line of
proof leading to the same conclusion. It had long been known
that the blue light of the sky, which is due to the scattering of
the sun's direct rays by small particles or molecules in the
-atmosphere, is partly polarized. The polarization is most
marked when the light comes from a part of the sky distant 90°
from the sun, in which case it must have been scattered in a
direction perpendicular to that of the direct sunlight incident
on the small particles ; and the polarization is in the plane
through the sun.
If, then, the axis of y be taken parallel to the light incident
on a small particle at the origin, and the scattered light be
observed along the axis of x, this scattered light is found to be
polarized in the plane xy. Considering the matter from the
dynamical point of view, we may suppose the material particle
to possess so much inertia (compared to the aether) that it is
practically at rest. Its motion relative to the aether, which is
the cause of the disturbance it creates in the aether, will there-
fore be in the same line as the incident aethereal vibration,
but in the opposite direction. The disturbance must be
transversal, and must therefore be zero in a polar direction and
* Ann. d. Phj-s. exi (1860), p. 315. Phil. Mag. xxi (1861), p. 321.
t Phil. Trans., 1852, p. 463. Stokes's Math, and Phys. Papers, iii, p. 267.
€f. the foot-note added on p. 361 oi the Math, and Phys. Paper*.
170 The Aether as an Elastic Solia.
a maximum in an equatorial direction, its amplitude being, in
fact, proportional to the sine of the polar distance. The polar
line must, by considerations of symmetry, be the line of the
incident vibration. Thus we see that none of the light scattered
in the ^-direction can come from that constituent of the incident
light which vibrates parallel to the o>axis ; so the light observed
in this direction must consist of vibrations parallel to the 2-axis.
But we have seen that the plane of polarization of the scattered
light is the plane of xy ; and therefore the vibration is at right
angles to the plane of polarization.*
The phenomena of diffraction and of polarization by scatter-
ing thus agreed in confirming the result arrived at in Fresnel's
and Green's theory of reflexion. The chief difficulty in accepting
it arose in connexion with the optics of crystals. As we have
seen, Green and Cauchy were unable to reconcile the hypothesis
of aethereal vibrations at right angles to the plane of polariza-
tion with the correct formulae of crystal-optics, at any rate so
long as the aether within crystals was supposed to be free from
initial stress. The underlying reason for this can be readily
seen. In a crystal, where the elasticity is different in different
directions, the resistance to distortion depends solely on the
orientation of the plane of distortion, which in the case of light
is the plane through the directions of propagation and vibration.
Now it is known that for light propagated parallel to one of the
axes of elasticity of a crystal, the velocity of propagation
depends only on the plane of polarization of the light, being the
same whichever of the two axes lying in that plane is the
direction of propagation. Comparing these results, we see that
the plane of polarization must be the plane of distortion, and
therefore the vibrations of the aether-particles must be executed
parallel to the plane of polarization.f
* The theory of polarization by small particles was afterwards investigated by
Lord Rayleigh, Phil. Mag. xli(187l).
fin Fresnel's theory of crystal-optics, in which the aether-vibrations are at
right angles to the plane of polarization, the velocity of propagation depends only
on the direction of vibration, not on the plane through this and the direction of
transmission.
The Aether as an Elastic Solid. 171
A way of escape from this conclusion suggested itself to
Stokes,* and later to Eankinet and Lord Kayleigh.J; What if the
aether in a crystal, instead of having its elasticity different in
different directions, were to have its rigidity invariable and its
inertia different in different directions ? This would bring the
theory of crystal- op tics into complete agreement with Fresnel's
and Green's theory of reflexion, in which the optical differences
between media are attributed to differences of inertia of the
aether contained within them. The only difficulty lies in
conceiving how aelotropy of inertia can exist; and all three
writers overcame this obstacle by pointing out that a solid
which is immersed in a fluid may have its effective inertia
different in different directions. For instance, a coin immersed
in water moves much more readily in its own plane than in the
direction at right angles to this.
Suppose then that twice the kinetic energy per unit volume
of the aether within a crystal is represented by the expression
and that the potential energy per unit volume has the same
value as in space void of ordinary matter. The aether is
assumed to be incompressible, so that div e is zero : the potential
energy per unit volume is therefore
_ __
dz dx d
where n denotes as usual the rigidity.
* Stokes, in a letter to Lord Rayleigh, inserted in his Memoir and Scientific
Correspondence, ii, p. 99, explains that the idea presented itself to him while he
was writing the paper on Fluid Motion which appeared in Trans. Camh. Phil.
Soc., via (1843), p. 105. He suggested the wave-surface to which this theory
leads in Brit. Assoc. Rep., 1862, p. 269.
t Phil. Mag. (4), i (1851), p. 441. J Phil. Mag. (4), xli (1871), p. 519.
1 72 The Aether as an Elastic Solid.
The variational equation of motion is
pi ^r/ dex + pz —% dey + pz — 2 $ez[ dx dy dz
where p denotes an undetermined function of (x, y, z) : the term
in p being introduced on account of the kinematical constraint
expressed by the equation
div e = 0.
The equations of motion which result from this variational
equation are
<hw=-£+nV*e" ••'""•'
and two similar equations. It is evident that p resembles a
hydrostatic pressure.
Substituting in these equations the analytical expression
for a plane wave, we readily find that the velocity F of the
wave is connected with the direction-cosines (X, ^t, z/) of its
normal by the equation
A2 u* vz
n- PIV* r n-ptV" + n- pzV* = '
When this is compared with Fresnel's relation between the
velocity and direction of a wave, it is seen that the new formula
differs from his only in having the reciprocal of the velocity in
place of the velocity. About 1867 Stokes carried out a series
of experiments in order to determine which of the two theories
was most nearly conformable to the facts : he found the con-
struction of Huygens and Fresnel to be decidedly the more
correct, the difference between the results of it and the rival
construction being about 100 times the probable error of
observation.*
* Proc. R. S., June, 1872. After these experiments Stokes gave it as his opinion
(Phil. Mag. xli (1871), p. 521) that the true theory of crystal-optics was yet to be
found. On the accuracy of Fresnel's construction cf. Glazebrook, Phil. Trans,
clxxi (1879) p. 421, and Hastings, Am. Journ. Sci. (3) xxxv (1887) p. 60.
The Aether as an Elastic Solid. 173
The hypothesis that in crystals the inertia depends on
direction seemed therefore to be discredited when the theory
based on it was compared with the results of observation. But
when, in 1888, W. Thomson (Lord Kelvin) revived Cauchy's
theory of the labile aether, the question naturally arose as to
whether that theory could be extended so as to account for the
optical properties of crystals : and it was shown by E. T.
Glazebrook* that the correct formulae of crystal-optics ar&
obtained when the Cauchy-Thomson hypothesis of zero velocity
for the longitudinal wave is combined with the Stokes-Kankine-
Rayleigh hypothesis of aelotropic inertia.
For on reference to the formulae which have been already
given, it is obvious that the equation of motion of an aether
having these properties must be
(pie*, pzey, p3O = -n curl curl e,
where e denotes the displacement, n the rigidity, and (plt p2, /o3)
the inertia : and this equation leads by the usual analysis ta
Fresnel's wave-surface. The displacement e of the aethereal
particles is not, however, accurately in the wave-front, as in
Fresnel's theory, but is at right angles to the direction of the
ray, in the plane passing through the ray and the wave-
normal, f
Having now traced the progress of the elastic-solid theory
so far as it is concerned with the propagation of light in
ordinary isotropic media and in crystals, we must consider the
attempts which were made about this time to account for the
optical properties of a more peculiar class of substances.
It was found by Arago in 181 IJ that the state of
polarization of a beam of light is altered when the beam is
passed through a plate of quartz along the optic axis. The
* Phil. Mag. xxvi (1888), p. 521 ; xxviii (1889), p. 110.
t This theory of crystal-optics may be assimilated to the electro-magnetic theory
by interpreting the elastic displacement e as electric force, and the vector
(pifx, p^y, ptfz) as electric displacement.
+ Mem. de 1'Institut, 1811, Part I, p. 115, sqq.
174 The Aether as an Elastic Solid.
phenomenon was studied shortly afterwards by Biot,* who
showed that the alteration consists in a rotation of the plane of
polarization about the direction of propagation : the angle of
rotation is proportional to the thickness of the plate and
inversely proportional to the square of the wave-length.
In some specimens of quartz the rotation is from left to
right, in others from right to left. This distinction was shown
by Sir John Herschelf (b. 1792, d. 1871) in 1820 to be
associated with differences in the crystalline form of the
specimens, the two types bearing the same relation to each
other as a right-handed and left-handed helix respectively.
FresnelJ and W. Thomsong proposed the term helical to
denote the property of rotating the plane of polarization,
exhibited by such bodies as quartz : the less appropriate term
natural rotatory polarization is, however, generally used.||
Biot showed that many liquid organic bodies, e.g. turpentine
and sugar solutions, possess the natural rotatory property : we
might be led to infer the presence of a helical structure in
the molecules of such substances ; and this inference is sup-
ported by the study of their chemical constitution; for they
are invariably of the "mirror-image" or "enantiomorphous"
type, in which one of the atoms (generally carbon) is asym-
metrically linked to other atoms.
The next advance in the subject was due to Fresnel,1[ who
showed that in naturally active bodies the velocity of propa-
gation of circularly polarized light is different according as the
polarization is right-handed or left-handed. From this
property the rotation of the plane of polarization of a plane-
polarized ray may be immediately deduced ; for the plane-
polarized ray may be resolved into two rays circularly polarized
in opposite senses, and these advance in phase by different
* Mem. de 1'Institut, 1812, Part i, p. 218, sqq. ; Annales de Chim., ix (1818),
p. 372; x (1819), p. 63. tCamb. Phil. Soc. Trans, i, p. 43.
J Mem. de 1'Inst. vii, p. 73. § Baltimore Lectures (ed. 1904), p. 31.
|| The term rotatory may be applied with propriety to the property discovered
by Faraday, which will be discussed later.
H Annales de Chim. xxviii (1825), p. 147.
The Aether as an Elastic Solid. 1 75
amounts in passing through a given thickness of the substance-:
at any stage they may be recompoundecl into a pkne-polarized
ray, the azimuth of whose plane of polarization varies with the
length of path traversed.
It is readily seen from this that a ray of light incident on
a crystal of quartz will in general bifurcate into two refracted
rays, each of which will be elliptically polarized, i.e. will be
capable of resolution into two plane-polarized components
which differ in phase by a definite amount. The directions of
these refracted rays may be determined by Huygens' con-
struction, provided the wave-surface is supposed to consist of a
sphere and spheroid which do not touch.
The first attempt to frame a theory of naturally active
bodies was made by MacCullagh in 1836.* Suppose a plane
wave of light to be propagated within a crystal of quartz. Let
(#, ?/, z) denote the coordinates of a vibrating molecule, when
the axis of x is taken at right angles to the plane of the wave,
and the axis of z at right angles to the axis of the
crystal. Using Fand Zto denote the displacements parallel to
the axes of y and z respectively at any time t, MacCullagh
assumed that the differential equations which determine Y and
__ _
w "" ** w ^'w
where /* denotes a constant on which the natural rotatory
property of the crystal depends. In order to avoid compli-
cations arising from the ordinary crystalline properties of quartz,
we shall suppose that the light is propagated parallel to the
optic axis, so that we can take c, equal to c2.
Assuming first that the beam is circularly polarized, let it
be represented by
(y f\
Y = A sin — (Ix - £), Z = ± A cos — (Ix - t),
* Trans. Royal Irish Acad., xvii. ; MacCullagh's Coll. Works, p. 63.
176 The Aether as an Elastic Solid.
*he ambiguous sign being determined according as the circular
polarization. ^ ri^ht-handed or left-handed.
Substituting in ^e above differential equations, we have
or
Since I// denotes the velocity of propagation, it is evident that
the reciprocals of the velocities of propagation of a right-handed
and left-handed beam differ by the quantity
from which it is easily shown that the angle through which the
plane of polarization of a plane-polarized beam rotates in unit
length of path is
rV
If we neglect the variation of Ci with the period of the light,
this expression satisfies Biot's law that the angle of rotation
in unit length of path is proportional to the inverse square of
the wave-length.
MacCullagh's investigation can be scarcely called a theory,
for it amounts only to a reduction of the phenomena to
empirical, though mathematical, laws ; but it was on this
foundation that later workers built the theory which is now
accepted.*
* The later developments of this theory will be discussed in a subsequent
chapter ; hut mention may here he made of an attempt which was made in 1856 by
Carl Neumann, then a very young man, to provide a rational basis for MacCullagh's
equations. Neumann showed that the equations may be derived from the
hypothesis that the relative displacement of one aethereal particle with respect to
another acts on the latter according to the same law as an element of an electric
current acts on a magnetic pole. Cf. the preface to C. Neumann's Die
Drehung der Polarisationsebene des Lichtes, Halle, 1863.
The Aether as an Elastic Solid. 177
The great investigators who developed the theory of light
after the death of Fresnel devoted considerable attention to
the optical properties of metals. Their researches in this
direction must now be reviewed.
The most striking properties of metals are the power of
brilliantly reflecting light at all angles of incidence, which is
so well shown by the mirrors of reflecting telescopes, and the
opacity, which causes a train of waves to be extinguished before
it has proceeded many wave-lengths into a metallic medium.
That these two attributes are connected appears probable
from the fact that certain non- metallic bodies — e.g., aniline
dyes — which strongly absorb the rays in certain parts of the
spectrum, reflect those rays with almost metallic brilliance.
A third quality in which metals differ from transparent bodies,
and which, as we shall see, is again closely related to the other
two, is in regard to the polarization of the light reflected from
them. This was first noticed by Malus ; and in 1830 Sir David
Brewster* showed that plane-polarized light incident on a
metallic surface remains polarized in the same plane after
reflexion if its polarization is either parallel or perpendicular
to the plane of reflexion, but that in other cases the reflected
light is polarized elliptically.
It was this discovery of Brews ter's which suggested to the
mathematicians a theory of metallic reflexion. For, as we have
seen, elliptic polarization is obtained when plane-polarized
light is totally reflected at the surface of a transparent body ;
and this analogy between the effects of total reflexion and
metallic reflexion led to the surmise that the latter pheno-
menon might be treated in the same way as Fresnel had treated
the former, namely, by introducing imaginary quantities into
the formulae of ordinary reflexion. On these principles mathe-
matical formulae were devised by MacCullaghf and Cauchy^
*Phil. Trans., 1830.
+ Proc. Roy. Irish Acad., i (1836), p. 2 ; ii (1843), p. 376 : Trans. Roy. Irish
Acad., xviii (1837), p. 71 : MacCullagh's Coll. Works, pp. 58, 132, 230.
J Comptes Rendus, vii (1838), p. 953 ; riii (1839), pp. 553, 658, 961 ; xxvi
(1848), p. 86.
N
178 The Aether as an Elastic Solid.
To explain their method, we shall suppose the incident
light to be polarized in the plane of incidence. According to
Fresnel's sine-law, the amplitude of the light (polarized in this
way) reflected from a transparent body is to the amplitude of
the incident light in the ratio
_ sin (i - r)
sin (i + r)'
where i denotes the angle of incidence and r is determined from
the equation
sin i = ft sin r.
MacCullagh and Cauchy assumed that these equations hold good
also for reflexion at a metallic surface, provided the refractive
index /* is replaced by a complex quantity
IJL = v(l — *v/ — 1) say,
where v and K are to be regarded as two constants characteristic
of the metal. We have therefore
tan i - tan r (ju2 - sin2 i)% - cos i
jj — ,.-,.. - — • - - — -- "
tan i + tan r (fj2 - sin2 i)k + cos i
If then we write
so that equations defining U and v are obtained by equating
separately the real and the imaginary parts of this equation, we
have
Ue^ ~ l - cos i
J
TT v\/ — 1
Ue v + cos
and this may be written in the form
where
-=.2 U* + cos2^ - 2 U cos v cos i
U" + cos2* + 2 U cos v cos i
2 U cos i sin v
tang =
U* - cos2*
The Aether as an Elastic Solid. 179
The quantities J and S are interpreted in the same way as
in Fresnel's theory of total reflexion : that is, we take J to
mean the ratio of the intensities of the reflected and incident
light, while 3 measures the change of phase experienced by
the light in reflexion.
The case of light polarized at right angles to the plane of
incidence may be treated in the same way.
When the incidence is perpendicular, U evidently reduces
to v (1 + K2)*, and u reduces to - tan-1 K. For silver at perpen-
— 2
dicular incidence almost all the light is reflected, so J is nearly
unity : this requires cos v to be small, and K to be very large.
The extreme case in which K is indefinitely great but v indefinitely
small, so that the quasi- index of refraction is a pure imaginary,
is generally known as the case of ideal silver.
The physical significance of the two constants v and K was
more or less distinctly indicated by Cauchy; in fact, as the
difference between metals and transparent bodies depends on
the constant K, it is evident that K must in some way measure
the opacity of the substance. This will be more clearly seen if
we inquire how the elastic-solid theory of light can be extended
so as to provide a physical basis for the formulae of MacCullagh
and Cauchy.* The sine-formula of Fresnel, which was the
starting-point of our investigation of metallic reflexion, is a
consequence of Green's elastic-solid theory : and the differences
between Green's results and those which we have derived arise
solely from the complex value which we have assumed for yu.
We have therefore to modify Green's theory in such a way as
to obtain a complex value for the index of refraction.
Take the plane of incidence as plane of xy, and the metallic
surface as plane of yz. If the light is polarized in the plane of
incidence, so that the light- vector is parallel to the axis of z,
the incident light may be taken to be a function of the
argument
ax + by + ct,
* This was done by Lord Rayleigh, Phil. Mag. xliii (1872), p. 321.
N 2
180 The Aether as an Elastic Solid.
where
a /p\l . b
- = - - COS I,
c \n c
/p\*
- - I sin ^ ;
\nj
here * denotes the angle of incidence, p the inertia of the aether,,
and n its rigidity.
Let the reflected light be a function of the argument
OiX + by + ct,
where, in order to secure continuity at the boundary, b and c
must have the same values as before. Since Green's formulae
are to be still applicable, we must have
where sin i = ft sin r, but /j. has now a complex value. This-
equation may be written in the form
n
Let the complex value of /u,z be written
p
the real part being written pi/p in order to exhibit the analogy
with Green's theory of transparent media : then we have
n n
But an equation of this kind must (as in Green's theory)
represent the condition to be satisfied in order that the
quantity
(a\x + by + ct) \/ - I
t/
may satisfy the differential equation of motion of the aether ;
from which we see that the equation of motion of the aether
in the metallic medium is probably of the form
dzez A dez
This equation of motion differs from that of a Greenian
The Aether as an Elastic Solid. 181
elastic solid by reason of the occurrence of the term in dez/dt.
But this is evidently a " viscous " term, representing something
like a frictional dissipation of the energy of luminous vibra-
tions : a dissipation which, in fact, occasions the opacity of the
metal. Thus the term which expresses opacity in the equation
of motion of the luminiferous medium appears as the origin of the
peculiarities of metallic reflexion.* It is curious to notice how
closely this accords with the idea of Huygens, that metals are
characterized by the presence of soft particles which damp the
vibrations of light.
There is, however, one great difficulty attending this
explanation of metallic reflexion, which was first pointed out by
Lord Rayleigh.f We have seen that for ideal silver ^ is real
and negative : and therefore A must be zero and p± negative ;
that is to say, the inertia of the luminiferous medium in the
metal must be negative. This seems to destroy entirely the
physical intelligibility of the theory as applied to the case of
ideal silver.
The difficulty is a deep-seated one, and was not overcome
for many years. The direction in which the true solution lies
will suggest itself when we consider the resemblance which
has already been noticed between metals and those substances
which show "surface colour" — e.g. the aniline dyes. In the
case of the latter substances, the light which is so copiously
reflected from them lies within a restricted part of the spectrum ;
and it therefore seems probable that the phenomenon is not
to be attributed to the existence of dissipative terms, but that
it belongs rather to the same class of effects as dispersion,
and is to be referred to the same causes. In fact, dispersion
means that the value of the refractive index of a substance
with respect to any kind of light depends on the period of
the light ; and we have only to suppose that the physical
causes which operate in dispersion cause the refractive index
* It is easily seen that the amplitude is reduced by the factor e-™* when light
travels one wave-length in the metal : K is generally called the coefficient of
absorption. 1" Loc. cit.
182 The Aether as an Elastic Solid.
to become imaginary for certain kinds of light, in order to
explain satisfactorily both the surface colours of the aniline
dyes and the strong reflecting powers of the metals.
Dispersion was the subject of several memoirs by the
founders of the elastic-solid theory. So early as 1830 Cauchy's
attention was directed* to the possibility of constructing a
mathematical theory of this phenomenon on the basis of
Fresnel's " Hypothesis of Finite Impacts "f — i.e. the assumption
that the radius of action of one particle of the luminiferous
medium on its neighbours is so large as to be comparable with
the wave-length of light. Cauchy supposed the medium to
be formed, as in Navier's theory of elastic solids, of a system
of point-centres of force : the force between two of these
point-centres, m at (x, y, z), and //, at (x + A#, y + Ay, z + Az),
may be denoted by m^/(r), where r denotes the distance between
m and p. When this medium is disturbed by light- waves pro-
pagated parallel to the z-axis, the displacement being parallel
to the #>axis, the equation of motion of m is evidently
-
+ p) -rT— ,
where £ denotes the displacement of m, (£ -i A If) the displace-
ment of p, and (r + p) the new value of r. Substituting for p its
value, and retaining only terms of the first degree in A?, this
equation becomes
DT r dr
Now, by Taylor's theorem, since £ depends only on z, we have
Substituting, and remembering that summations which
involve odd powers of Az must vanish when taken over all
* Bull, des Sc. Math, xiv (1830), p. 9 : " Sur la dispersion de la lumiere,"
. Exevcwe* de Math., 1836. t Cf. p. 132.
The Aether as an Elastic Solid. 183
the point-centres within the sphere of influence of m, we obtain
an equation of the form
fft d'K o^E d'Z
w = a&+Pw + y& + ---'
where a, )3, 7 . . . denote constants.
Each successive term on the right-hand side of this equation
involves an additional factor (A^)2/X8 as compared with the pre-
ceding term, where X denotes the wave-length of the light : so
if the radii of influence of the point-centres were indefinitely
small in comparison with the wave-length of the light, the
equation would reduce to
8^_ cP£
ar- = "&*'
which is the ordinary equation of wave-propagation in one
dimension in non-dispersive media. But if the medium is so
coarse-grained that A is not large compared with the radii of
influence, we must retain the higher derivates of £. Substi-
tuting
in the differential equation with these higher derivates retained,
we have
'2-irV
which shows that cb the velocity of the light in the medium,
depends on the wave-length A ; as it should do in order to
explain dispersion.
Dispersion is, then, according to the view of Fresnel and
Cauchy, a consequence of the coarse-grainedness of the medium.
Since the luminiferous medium was found to be dispersive only
within material bodies, it seemed natural to suppose that in
these bodies the aether is loaded by the molecules of matter,
and that dispersion depends essentially on the ratio of the
wave-length to the distance between adjacent material molecules.
184 The Aether as an Elastic Solid.
This theory, in one modification or another, held its ground
until forty years later it was overthrown by the facts of
anomalous dispersion.
The distinction between aether and ponderable matter was
more definitely drawn in memoirs which were published
independently in 1841-2 by F. E. Neumann* and Matthew
O'Brien.f These authors supposed the ponderable particles to
remain sensibly at rest while the aether surges round them, and
is acted on by them with forces which are proportional to its
displacement. ThusJ the equation of motion of the aether
becomes
rP&
p •£ ~2 = - (k + ^n) grad div e - n curl curl e - Ce,
ot
where C denotes a constant on which the phenomena of dis-
persion depend. For polarized plane waves propagated parallel
to the axis of x, this equation becomes
92e 92e „
fg^»5r*5
and substituting
e = e
where r denotes the period and V the velocity of the light, we
have
G T,
772 - P 4^3 r '
an equation which expresses the dependence of the velocity on
the period.
The attempt to represent the properties of the aether by
those of an elastic solid lost some of its interest after the
rise of the electromagnetic theory of light. But in 1867,
* Berlin Abhandlungen aus dem Jahre 1841, Zweiter Teil, p. 1 : Berlin, 1843.
t Trans. Camb. Phil. Soc. vii (1842), p. 397.
J O'Brien, loc. cit, §§ 15, 28.
The Aether as an Elastic Solid. 185
before the electromagnetic hypothesis had attracted much
attention, an elastic-solid theory in many respects preferable
to its predecessors was presented to the French Academy* by
Joseph Boussinesq (b. 1842). Until this time, as we have
seen, investigators had been divided into two parties, according
as they attributed the optical properties of different bodies to
variations in the inertia of the luminiferous medium, or to
variations in its elastic properties. Boussinesq, taking up a
position apart from both these schools, assumed that the aether
is exactly the same in all material bodies as in interplanetary
space, in regard both to inertia and to rigidity, and that the
optical properties of matter are due to interaction between the
aether and the material particles, as had been imagined more or
less by Neumann and O'Brien. These material particles he
supposed to be disseminated in the aether, in much the same
way as dust-particles floating in the air.
If e denote the displacement at the point (x, y, z) in the
aether, and e' the displacement of the ponderable particles
at the same place, the equation of motion of the aether is
rfie ?P&'
P *jp = ~ (k + ^l) g11"1 div e + ^V2e - p, jp, (1)
where p and pl denote the densities of the aether and matter
respectively, and k and n denote as usual the elastic constants
of the aether. This differs from the ordinary Cauchy-Green
equation only in the presence of the term pi&*'/dP, which
represents the effect of the inertia of the matter. To this
equation we must adjoin another expressing the connexion
between the displacements of the matter and of the aether:
if we assume that these are simply proportional to each
other — say,
e' = Ae, (2)
* Journal de Math. (2) xiii (1868), pp. 313, 425 : cf. also Comptes Rendus,
•cxvii (1893), pp. 80, 139, 193. Equations kindred to some of those of Boussinesq
M-ere afterwards deduced by Karl Pearson, Proc. Lond. Math. Soc , xx (1889),
p. 297, from the hypothesis that the strain-energy involves the velocities.
186 The Aether as an Elastic Solid.
where the constant A depends on the nature of the ponderable
body — our equation becomes
32e
(p + P1A) ^ = - (k + Jw) grad div e + ^V2e,
ot
which is essentially the same equation as is obtained in those
older theories which suppose the inertia of the luminiferous
medium to vary from one medium to another. So far there
would seem to be nothing very new in Boussinesq's work. But
when we proceed to consider crystal-optics, dispersion, and
rotatory polarization, the advantage of his method becomes
evident: he retains equation (1) as a formula universally true —
at any rate for bodies at rest — while equation (2) is varied
to suit the circumstances of the case. Thus dispersion can be
explained if, instead of equation (2), we take the relation
e' = Ae - Z>V2e,
where D is a constant which measures the dispersive power of
the substance : the rotation of the plane of polarization of sugar
solutions can be explained if we suppose that in these bodies
equation (2) is replaced by
e' = AQ + B curl e,
where B is a constant which measures the rotatory power ; and
the optical properties of crystals can be explained if we suppose
that for them equation (2) is to be replaced by the equations
ex' = Atfx, ey = Azeyt ez' = A3e,
When these values for the components of e' are substituted
in equation (1), we evidently obtain the same formulae as were
derived from the Stokes-Eankine-Eayleigh hypothesis of inertia
different in different directions in a crystal; to which Boussinesq's
theory of crystal-optics is practically equivalent.
The optical properties of bodies in motion may be accounted
for by modifying equation (1), so that it takes the form
a a a ay ,
- + Wx— + Wy—- + W~ — C ,,
ct cv oy ozj
The Aether as an Elastic Solid. 187
where w denotes the velocity of the ponderable body. If the
body is an ordinary isotropic one, and if we consider light
propagated parallel to the axis of z, in a medium moving in
that direction, the light- vector being parallel to the axis of x,
the equation reduces to
d'ex d'ex id 9V
O — 7b ' — Q]A. I -f- IV — I 6r i
' O/2 ^W- • \ ^/ ^i/v /
C7t (j6 \(7£ (72'/
substituting
«,-/(*- FO,
where V denotes the velocity of propagation of light in the
medium estimated with reference to the fixed aether, we obtain
for V the value
/ n \k o\A
\p + pt p +
The absolute velocity of light is therefore increased by the
amount piAw/(p + piA) owing to the motion of the medium ;
and this may be written (/** - 1) wjfjc, where ju denotes the
refractive index ; so that Boussinesq's theory leads to the same
formula as had been given half a century previously by Fresnel.*
It is Boussinesq's merit to have clearly asserted that all
space, both within and without ponderable bodies, is occupied
by one identical aether, the same everywhere both in inertia
and elasticity; and that all aethereal processes are to be re-
presented by two kinds of equations, of which one kind expresses
the invariable equations of motion of the aether, while the other
kind expresses the interaction between aether and matter.
Many years afterwards these ideas were revived in connexion
with the electromagnetic theory, in the modern forms of which
they are indeed of fundamental importance.
* Cf. p. 115 sqq.
( 188 )
CHAPTEK VI.
FAKADAY.
TOWARDS the end of the year 1812, Davy received a letter in
which the writer, a bookbinder's journeyman named Michael
Faraday, expressed a desire to escape from trade, and obtain
employment in a scientific laboratory. With the letter was
enclosed a neatly written copy of notes which the young man
— he was twenty-one years of age — had made of Davy's own
public lectures. The great chemist replied courteously, and
arranged an interview ; at which he learnt that his correspon-
dent had educated himself by reading the volumes which came
into his hands for binding. "There were two," Faraday
wrote later, "that especially helped me, the 'Encyclopaedia
Britannica,' from which I gained my first notions of electricity,
and Mrs. Marcet's ' Conversations on Chemistry/ which gave
me my foundation in that science." Already, before his applica-
tion to Davy, he had performed a number of chemical
experiments, and had made for himself a voltaic pile, with
which he had decomposed several compound bodies.
At Davy's recommendation Faraday was in the following
spring appointed to a post in the laboratory of the Koyal
Institution, which had been established at the close of the
eighteenth century under the auspices of Count Rumford ; and
here he remained for the whole of his active life, first as
assistant, then as director of the laboratory, and from 1833
onwards as the occupant of a chair of chemistry which was
founded for his benefit.
For many years Faraday was directly under Davy's influence,
and was occupied chiefly in chemical investigations. But in
1821, when the new field of inquiry opened by Oersted's
Faraday. 189
discovery was attracting attention, he wrote an Historical
Sketch of Electro- Magnetism* as a preparation for which he
carefully repeated the experiments described by the writers he
was reviewing ; and this seems to have been the beginning of
the researches to which his fame is chiefly due.
The memoir which stands first in the published volumes of
Faraday's electrical workf was communicated to the Royal
Society on November 24th, 1831. The investigation was
inspired, as he tells us, by the hope of discovering analogies
between the behaviour of electricity as observed in motion in
currents, and the behaviour of electricity at rest on conductors.
Static electricity was known to possess the power of " induction "
— i.e., of causing an opposite electrical state on bodies in its
neighbourhood ; was it not possible that electric currents might
show a similar property ? The idea at first was that if in any
circuit a current were made to flow, any adjacent circuit would
be traversed by an induced current, which would persist exactly
as long as the inducing current. Faraday found that this was
not the case ; a current was indeed induced, but it lasted only
for an instant, being in fact perceived only when the primary
current was started or stopped. It depended, as he soon
convinced himself, not on the mere existence of the inducing
current, but on its variation.
Faraday now set himself to determine the laws of induction
of currents, and for this purpose devised a new way of repre-
senting the state of a magnetic field. Philosophers had been
long accustomed? to illustrate magnetic power by strewing iron
filings on a sheet of paper, and observing the curves in which
they dispose themselves when a magnet is brought underneath.
•Published in Annals of Philosophy, ii (1821), pp. 195, 274; iii (1822),
p. 107.
t Experimental Researches in Electricity, by Michael Faraday : 3 vols.
* The practice goes back at least as far as Niccolo Cabeo ; indeed the curves
traced by Petrus Peregrinus on his globular lodestone (cf . p. 8) were projections
of lines of force. Among eighteenth-century writers La Hire mentions the use of
iron filings, Mem. de 1'Acad., 1717. Faraday had referred to them in his electro-
magnetic paper of 1821, Exp. Res. ii, p. 127.
190 Faraday.
These curves suggested to Faraday* the idea of lines of magnetic
force, or curves whose direction at every point coincides with
the direction of the magnetic intensity at that point; the
curves in which the iron filings arrange themselves on the
paper resemble these curves so far as is possible subject to the
condition of not leaving the plane of the paper.
With these lines of magnetic force Faraday conceived all
space to be filled. Every line of force is a closed curve, which
in some part of its course passes through the magnet to which
it belongs, f Hence if any small closed curve be taken in space,
the lines of force which intersect this curve must form a
tubular surface returning into itself ; such a surface is called a
tiibe of force. From a tube of force we may derive information
not only regarding the direction of the magnetic intensity,
but also regarding its magnitude; for the product of this
magnitude} and the cross-section of any tube is constant along
the entire length of the tube.§ On the basis of this result,
Faraday conceived the idea of partitioning all space into
compartments by tubes, each tube being such that this product
has the same definite value. For simplicity, each of these
tubes may be called a " unit line of force " ; the strength of
the field is then indicated by the separation or concentration of
the unit lines of force,! I so that the number of them which
intersect a unit area placed at right angles to their direction
#They were first defined in Exp. Res., § 114 : "By magnetic curves, I mean
the lines of magnetic forces, however modified hy the juxtaposition of poles,
which could be depicted by iron filings ; or those to which a very small magnetic
needle would form a tangent."
t Exp. Res. iii, p. 405.
J Within the substance of magnetized bodies we must in this connexion under-
stand the magnetic intensity to be that experienced in a crevice whose sides are
perpendicular to the lines of magnetization : in other words, we must take it to be
what since Maxwell's time has been called the magnetic induction.
§ Exp. Res., § 3073. This theorem was first proved by the French geometer
Michel Chasles, in his memoir on the attraction of an ellipsoidal sheet, Journal
de 1'Ecole Polyt. xv (1837), p. 266.
|| Ibid., § 3122. "The relative amount of force, or of lines of force, in a
given space is indicated by their concentration or separation — i.e., by their number
in that space."
Faraday. 191
at any point measures the intensity of the magnetic field at
that point.
Faraday constantly thought in terms of lines of force.
" I cannot refrain," he wrote, in 1851,* " from again expressing
my conviction of the truthfulness of the representation, which
the idea of lines of force affords in regard to magnetic action.
All the points which are experimentally established in regard
to that action — i.e. all that is not hypothetical — appear to be well
and truly represented by it."f
Faraday found that a current is induced in a circuit either
when the strength of an adjacent current is altered, or when a
magnet is brought near to the circuit, or when the circuit itself
is moved about in presence of another current or a magnet.
He saw from the firstj that in all cases the induction depends
on the relative motion of the circuit and the lines of magnetic
force in its vicinity. The precise nature of this dependence
was the subject of long-continued further experiments. In
1832 he found§ that the currents produced by induction under
the same circumstances in different wires are proportional to
the conducting powers of the wires — a result which showed
that the induction consists in the production of a definite
electromotive force, independent of the nature of the wire, and
dependent only on the intersections of the wire and the
magnetic curves. This electromotive force is produced whether
the wire forms a closed circuit (so that a current flows) or is
open (so that electric tension results).
All that now remained was to inquire in what way the
electromotive force depends on the relative motion of the wire
and the lines of force. The answer to this inquiry is, in
* Exp. Res., § 3174.
t Some of Faraday's most distinguished contemporaries were far from sharing
this conviction. " I declare," wrote Sir George Airy in 1855, " that I can hardly
imagine anyone who practically and numerically knows this agreement " between
observation and the results of calculation based on action at a distance, "to hesitate
au instant in the choice between this simple and precise action, on the one hand,
and anything so vague and varying as lines of force, on the other hand." Cf.
Bence Jones's Life of Faraday, ii, p. 353.
I Exp. Res., § 116. § Ibid., § 213.
192 Faraday.
Faraday's own words,* that "whether the wire moves directly or
obliquely across the lines of force, in one direction or another, it
sums up the amount of the forces represented by the lines it
has crossed," so that " the quantity of electricity thrown into a
current is directly as the number of curves intersected."t The
induced electromotive force is, in fact, simply proportional to
the number of the unit lines of magnetic force intersected by
the wire per second.
This is the fundamental principle of the induction of
currents. Faraday is undoubtedly entitled to the full honour
of its discovery ; but for a right understanding of the progress
of electrical theory at this period, it is necessary to remember
that many years elapsed before all the conceptions involved in
Faraday's principle became clear and familiar to his contem-
poraries ; and that in the meantime the problem of formulating
the laws of induced currents was approached with success from
other points of view. There were indeed many obstacles to the
direct appropriation of Faraday's work by the mathematical
physicists of his own generation ; not being himself a mathe-
matician, he was unable to address them in their own language ;
and his favourite mode of representation by moving lines of
force repelled analysts who had been trained in the school of
Laplace and Poisson. Moreover, the idea of electromotive force
itself, which had been applied to currents a few years previously
in Ohm's memoir, was, as we have seen, still involved in
obscurity and misapprehension.
A curious question which arose out of Faraday's theory
was whether a bar-magnet which is rotated on its own axis
carries its lines of magnetic force in rotation with it. Faraday
himself believed that the lines of force do not rotate J: on this
view a revolving magnet like the earth is to be regarded as
moving through its own lines of force, so that it must become
charged at the equator and poles with electricity of opposite
signs ; and if a wire not partaking in the earth's rotation were
to have sliding contact with the earth at a pole and at the
* Exp. Res., § 3082. t Ibid., § 3115. % Ibid., § 3090.
Faraday. 193
equator, a current would steadily flow through it. Experiments
confirmatory of these views were made by Faraday himself ;*
but they do not strictly prove his hypothesis that the lines of
force remain at rest ; for it is easily seenf that, if they were to
rotate, that part of the electromotive force which would be
produced by. their rotation would be derivable from a potential,
and so would produce no effect in closed circuits such as Faraday
used.
Three years after the commencement of Faraday's researches
on induced currents he was led to an important extension of
them by an observation which was communicated to him by
another worker. William Jenkin had noticed that an
electric shock may be obtained with no more powerful source of
electricity than a single cell, provided the wire through which
the current passes is long and coiled ; the shock being felt when
contact is broken. J As Jenkin did not choose to investigate
the matter further, Faraday took it up, and showed§ that the
powerful momentary current, which was observed when the
circuit was interrupted, was really an induced current governed
by the same laws as all other induced currents, but with this
peculiarity, that the induced and inducing currents now flowed
in the same circuit. In fact, the current in its steady state
establishes in the surrounding region a magnetic field, whose
lines of force are linked with the circuit ; and the removal of
these lines of force when the circuit is broken originates an
induced current, which greatly reinforces the primary current
just before its final extinction. To this phenomenon the name
of self-induction has been given.
The circumstances attending the discovery of self-induction
•Exp. Res., §$ 218, 3109, &c.
t Cf. W. Weber, Ann. d. Phys. lii (1841) ; S. Tolver Preston, Phil. Mag. xix
(1885), p. 131. In 1891 S. T. Preston, Phil. Mag. xxxi, p. 100, designed a crucial
experiment to test the question ; but it was not tried for want of a sufficiently
delicate electrometer.
% A similar observation had been made by Henry, and published in the Amer.
Jour. Sci. xxii (1832), p. 408. The spark at the rupture of a spirally-wound
circuit had been often observed, e.g., by Pouillet and Nobili.
§ Exp. Res., § 1048.
O
1 94 Faraday.
occasioned a comment from Faraday on the number of sugges-
tions which were continually being laid before him. He re-
marked that although at different times a large number of
authors had presented him with their ideas, this case of
Jenkin was the only one in which any result had followed.
" The volunteers are serious embarrassments generally to the
experienced philosopher."*
The discoveries of Oersted, Ampere, and Faraday had shown
the close connexion of magnetic with electric science. But the
connexion of the different branches of electric science with
each other was still not altogether clear. Although Wollaston's
experiments of 1801 had in effect proved the identity in kind
of the currents derived from frictional and voltaic sources, the
question was still regarded as open thirty years afterwards,f no
satisfactory explanation being forthcoming of the fact that
frictional electricity appeared to be a surface-phenomenon,
whereas voltaic electricity was conducted within the interior
substance of bodies. To this question Faraday now applied him-
self; and in 1833 he succeeded* in showing that every known
effect of electricity — physiological, magnetic, luminous, calorific,
chemical, and mechanical — may be obtained indifferently either
with the electricity which is obtained by friction or with that
obtained from a voltaic battery. Henceforth the identity of the
two was beyond dispute.
Some misapprehension, however, has existed among later
writers as to the conclusions which may be drawn from this
identification. What Faraday proved is that the process which
goes on in a wire connecting the terminals of a voltaic cell is of
the same nature as the process which for a short time goes on in
a wire by which a condenser is discharged. He did not prove,
* Bence Jones's Life of Faraday, ii, p. 45.
t Cf. John Davy, Phil. Trans., 1832, p. 259 ; W. Ritchie, ibid., p. 279. Davy
suggested that the electrical power, " according to the analogy of the solar ray,"
might be " not a simple power, but a combination of powers, which may occur
variously associated, and produce all the varieties of electricity with which we are
acquainted."
J Exp. Jies.9 Series iii.
Faraday. 195
and did not profess to have proved, that this process consists in
the actual movement of a quasi-substance, electricity, from one
plate of the condenser to the other, or of two quasi-substances,
the resinous and vitreous electricities, in opposite directions.
The process had been pictured in this way by many of his
predecessors, notably by Volta; and it has since been so
pictured by most of his successors : but from such assumptions
Faraday himself carefully abstained.
What is common to all theories, and is universally conceded,
is that the rate of increase in the total quantity of electrostatic
charge within any volume-element is equal to the excess of the
influx over the efflux of current from it. This statement may
be represented by the equation
|+divi = 0, (1)
where p denotes the volume-density of electrostatic charge,
and i the current, at the place (x, y, z) at the time t. Volta's
assumption is really one way of interpreting this equation
physically: it presents itself when we compare equation (1)
with the equation
which is the equation of continuity for a fluid of density p and
velocity v : we may identify the two equations by supposing i
to be of the same physical nature as the product /»v; and
this is precisely what is done by those who accept Volta's
assumption.
But other assumptions might be made which would equally
well furnish physical interpretations to equation (1). For
instance, if we suppose p to be the convergence of any vector of
which i is the time-flux,* equation (1) is satisfied automatically ;
* In symbols,
div 8 = - ,
where s denotes the vector in question.
02
196 Faraday.
we can picture this vector as being of the nature of a displace-
ment. By such an assumption we should avoid altogether the
necessity for regarding the conduction-current as an actual
flow of electric charges, or for speculating whether the drifting
charges are positive or negative ; and there would be no longer
anything surprising in the production of a null effect by the
coalescence of electric charges of opposite signs.
Faraday himself wished to leave the matter open, and to
avoid any definite assumption.* Perhaps the best indication of
his views is afforded by a laboratory notej- of date 1837 : —
"After much consideration of the manner in which the
electric forces are arranged in the various phenomena generally,.
I have come to certain conclusions which I will endeavour to
note down without committing myself to any opinion as to the
cause of electricity, i.e., as to the nature of the power. If
electricity exist independently of matter, then I think that the
hypothesis of one fluid will not stand against that of two fluids.
There are, I think, evidently what I may call two elements of
power, of equal force and acting toward each other. But these
powers may be distinguished only by direction, and may be no
more separate than the north and south forces in the elements
of a magnetic needle. They may be the polar points of the
forces originally placed in the particles of matter."
It may be remarked that since the rise of the mathematical
theory of electrostatics, the controversy between the supporters
of the one-fluid and the two-fluid theories had become
manifestly barren. The analytical equations, in which
interest was now largely centred, could be interpreted equally
well on either hypothesis; and there seemed to be little
prospect of discriminating between them by any new experi-
mental discovery. But a problem does not lose its fascination
*"His principal aim," said Helmholtz in the Faraday Lecture of 1881,
" was to express in his new conceptions only facts, with the least possible use of
hypothetical substances and forces. This was really a progress in general
scientific method, destined to purify science from the last remains of meta-
physics."
t Bence Jones's Life of Foradny^ ii, p. 77.
Faraday. 197
because it appears insoluble. " I said once to Faraday," wrote
Stokes to his father-in-law in 1879, " as I sat beside him at a
British Association dinner, that I thought a great step would
be made when we should be able to say of electricity that
which we say of light, in saying that it consists of undula-
tions. He said to me he thought we were a long way off that
yet."*
For his next series of researches,! Faraday reverted to
subjects which had been among the first to attract him as an
apprentice attending Davy's lectures : the voltaic pile, and the
relations of electricity to chemistry.
It was at this time generally supposed that the decomposi-
tion of a solution, through which an electric current is passed,
is due primarily to attractive and repellent forces exercised on
its molecules by the metallic terminals at which the current
enters and leaves the solution. Such forces had been assumed
both in the hypothesis of Grothuss and Davy, and in the rival
hypothesis of De La Eive ;+ the chief difference between these
being that whereas Grothuss and Davy supposed a chain 'of
decompositions and recompositions in the liquid, De La Rive
supposed the molecules adjacent to the terminals to be the
only ones decomposed, and attributed to their fragments the
power of travelling through the liquid from one terminal to the
other.
To test this doctrine of the influence of terminals, Faraday
moistened a piece of paper in a saline solution, and supported
it in the air on wax, so as to occupy part of the interval
between two needle-points which were connected with an
electric machine. When the machine was worked, the current
was conveyed between the needle-points by way of the
moistened paper and the two air-intervals on either side of it ;
and under these circumstances it was found that the salt under-
went decomposition. Since in this case no metallic terminals of
.any kind were in contact with the solution, it was evident that
* Stokes's Scientific Correspondence, vol. i, p. 353.
t Exp. Res., § 450 (1833). £ Cf. pp. 78-9.
198 Faraday.
all hypotheses which attributed decomposition to the action of
the terminals were untenable.
The ground being thus cleared by the demolition of previous
theories, Faraday was at liberty to construct a theory of his
own. He retained one of the ideas of Grothuss' and Davy's
doctrine, namely, that a chain of decompositions and recombi-
nations takes place in the liquid ; but these molecular processes
he attributed not to any action of the terminals, but to a power
possessed by the electric current itself, at all places in its
course through the solution. If as an example we consider
neighbouring molecules A, B, C, D, . . . of the compound — say
water, which was at that time believed to be directly decom-
posed by the current — Faraday supposed that before the
passage of the current the hydrogen of A would be in close
union with the oxygen of A, and also in a less close relation with
the oxygen atoms of B, C, D, . . . : these latter relations being
conjectured to be the cause of the attraction of aggregation in
solids and fluids.* When an electric current is sent through the
liquid, the affinity of the hydrogen of A for the oxygen of B is
strengthened, if A and B lie along the direction of the current ;
while the hydrogen of A withdraws some of its bonds from the
oxygen of A, with which it is at the moment combined. So
long as the hydrogen and oxygen of A remain in association,
the state thus induced is merely one of polarization ; but the
compound molecule is unable to stand the strain thus imposed
on it, and the hydrogen and oxygen of A part company from
each other. Thus decompositions take place, followed by
recombinations : with the result that after each exchange an
oxygen atom associates itself with a partner nearer to the
positive terminal, while a hydrogen atom associates with a
partner nearer to the negative terminal.
This theory explains why, in all ordinary cases, the evolved
substances appear only at the terminals ; for the terminals are
the limiting surfaces of the decomposing substance ; and, except
at them, every particle finds other particles having a contrary
*Exp. lies., §523.
Faraday. 199
tendency with which it can combine. It also explains why, in
numerous cases, the atoms of the evolved substances are not
retained by the terminals (an obvious difficulty in the way of
all theories which suppose the terminals to attract the atoms) :
for the evolved substances v are expelled from the liquid, not
drawn out by an attraction.
Many of the perplexities which had harassed the older
theories were at once removed when the phenomena were re-
garded from Faraday's point of view. Thus, mere mixtures (as
opposed to chemical compounds) are not separated into their
constituents by the electric current ; although there would seem
to be no reason why the Grothuss-Davy polar attraction should
not operate as well on elements contained in mixtures as on
elements contained in compounds.
In the latter part of the same year (1833) Faraday took up
the subject again.* It was at this time that he introduced the
terms which have ever since been generally used to describe
the phenomena of electro-chemical decomposition. To the
terminals by which the electric current passes into or out of the
decomposing body he gave the name electrodes. The electrode
of high potential, at which oxygen, chlorine, acids, &c., are
evolved, he called the anode, and the electrode of low potential,
at which metals, alkalis, and bases are evolved, the cathode.
Those bodies which are decomposed directly by the current
he named electrolytes ; the parts into which they are decomposed,
ions ; the acid ions, which travel to the anode, he named anions ;
and the metallic ions, which pass to the cathode, cations.
Faraday now proceeded to test the truth of a supposition
which he had published rather more than a year previously ,f
and which indeed had apparently been suspected by Gay-Lussac
and ThenardJ so early as 1811; namely, that the rate at which
an electrolyte is decomposed depends solely on the intensity of
the electric current passing through it, and not at all on the
size of the electrodes or the strength of the solution. Having
* Exp. Res., § 661. f /*'*., § 377 (Dec. 1832).
£ Recherches physico-chimiqucs faites sur la pile ; Paris, 1811, p. 12.
200 Faraday.
established the accuracy of this law,* he found by a comparison
of different electrolytes that the mass of any ion liberated by
a given quantity of electricity is proportional to its chemical
equivalent, i.e. to the amount of it required to combine with
some standard mass of some standard element. If an element
is %-valent, so that one of its atoms can hold in combination
n atoms of hydrogen, the chemical equivalent of this element
may be taken to be 1/n of its atomic weight ; and therefore
Faraday's result may be expressed by saying that an electric
current will liberate exactly one atom of the element in
question in the time which it would take to liberate n atoms
of hydrogen.-)-
The quantitative law seemed to Faraday:}: to indicate that
" the atoms of matter are in some way endowed or associated
with electrical powers, to which they owe their most striking
qualities, and amongst them their mutual chemical affinity."
Looking at the facts of electrolytic decomposition from this
point of view, he showed how natural it is to suppose that
the electricity which passes through the electrolyte is the exact
equivalent of that which is possessed by the atoms separated at
the electrodes ; which implies that there is a certain absolute
quantity of the electric power associated with each atom of
matter.
The claims of this splendid speculation he advocated with
conviction. " The harmony," he wrote, § " which it introduces
into the associated theories of definite proportions and electro-
chemical affinity is very great. According to it, the equivalent
weights of bodies are simply those quantities of them which
contain equal quantities of electricity, or have naturally equal
electric powers ; it being the ELECTRICITY which determines the
equivalent number, because it determines the combining force.
Or, if we adopt the atomic theory or phraseology, then the
*Exp. Res., §§ 713-821.
t In the modern units, 96580 coulombs of electricity must pass round the
circuit in order to liberate of each ion a number of grams equal to the quotient of
the atomic weight by the valency.
J Exp. Res., § 852. § Ibid., § 869.
Faraday. 201
atoms of bodies which are equivalent to each other in their
ordinary chemical action, have equal quantities of electricity
naturally associated with them. " But," he added, " I must
confess I am jealous of the term atom : for though it is very
easy to talk of atoms, it is very difficult to form a clear idea
of their nature, especially when compound bodies are under
consideration."
These discoveries and ideas tended to confirm Faraday in
preferring, among the rival theories of the voltaic cell, that one
to which all his antecedents and connexions predisposed him.
The controversy between the supporters of Volta's contact
hypothesis on the one hand, and the chemical hypothesis of
Davy and Wollaston on the other, had now been carried on
for a generation without any very decisive result. In Germany
and Italy the contact explanation was generally accepted, under
the influence of Christian Heiririch Pfaff, of Kiel (b. 1773,
d. 1852), and of Ohm, and, among the younger men, of Gustav
Theodor Fechner (b. 1801, d. 1887), of Leipzig,* and Stefano
Marianini (b. 1790, d. 1866), of Modena. Among French writers
De La Eive, of Geneva, was, as we have seen, active in support
of the chemical hypothesis; and this side in the dispute had
always been favoured by the English philosophers.
There is no doubt that when two different metals are put
in contact, a difference of potential is set up between them
without any apparent chemical action ; but while the contact
party regarded this as a direct manifestation of a "contact-
force " distinct in kind from all other known forces of nature,
* Johaim Christian Poggendorff (b. 1796, d. 1877), of Berlin, for long the editor
of the Annalen der Physik, leaned originally to the chemical side, but in 1838
became convinced of the truth of the contact theory, which he afterwards actively
defended. Moritz Hermann Jacobi (b. 1801, d. 1874), of Dorpat, is also to be
mentioned among its advocates.
Faraday's first series of investigations on this subject were made in 1834 :
Exp, £es., series viii. In 1836 De La Kive followed on the same side with his
Eccherches sur la Cause de V Electr. Voltaique. The views of Faraday and De La
Rive were criticized by Pfaff, Revision der Lehre vom Galvanistntts, Kiel, 1837, and
by Fechner, Ann. d. Phys., xlii (1837), p. 481, and xliii (1838), p. 433 : translated
Phil. Mag., xiii (1838), pp. 205, 367. Faraday returned to the question in 1840,
Exp, Jtes., series xvi and xvii.
202 Faraday.
the chemical party explained it as a consequence of chemical
affinity or incipient chemical action between the metals and
the surrounding air or moisture. There is also no doubt that
the continued activity of a voltaic cell is always accompanied
by chemical unions or decompositions ; but while the chemical
party asserted that these constitute the efficient source of the-
current, the contact party regarded them as secondary actions,
and attributed the continual circulation of electricity to the
perpetual tendency of the electromotive force of contact to
transfer charge from one substance to another.
One of the most active supporters of the chemical theory
among the English physicists immediately preceding Faraday
was Peter Mark Eoget (b. 1779, d. 1869), to whom are due two-
of the strongest arguments in its favour. In the first place,
carefully distinguishing between the quantity of electricity put
into circulation by a cell and the tension at which this electricity
is furnished, he showed that the latter quantity depends on the
" energy of the chemical action "* — a fact which, when taken
together with Faraday's discovery that the quantity of electricity
put into circulation depends on the amount of chemicals con-
sumed, places the origin of voltaic activity beyond all question.
Koget's principle was afterwards verified by Faradayf and by
De La EiveJ; " the electricity of the voltaic pile is proportionate
in its intensity to the intensity of the affinities concerned in
its production," said the former in 1834; while De La Kive
wrote in 1836, " The intensity of the currents developed in
combinations and in decompositions is exactly proportional to
the degree of affinity which subsists between the atoms whose
combination or separation has given rise to these currents."
* " The absolute quantity of electricity which is thus developed, and made to
circulate, will depend upon a variety of circumstances, such as the extent of the
surfaces in chemical action, the facilities afforded to its transmission, &c. But
its degree of intensity, or tension, as it is often termed, will be regulated by other
causes, and more especially by the energy of the chemical action." Roget's
Galvanism (1832), § 70.
t Exp. Res., §§ 908, 909, 916, 988, 1958.
I Annales de Chim., Ixi (1836), p. 38.
Faraday. 203
Not resting here, however, Koget brought up another argu-
ment of far-reaching significance. " If," he wrote,* " there could
exist a power having the property ascribed to it by the [contact]
hypothesis, namely, that of giving continual impulse to a fluid
in one constant direction, without being exhausted by its own
action, it would differ essentially from all the other known
powers in nature. All the powers and sources of motion, with
the operation of which we are acquainted, when producing their
peculiar effects, are expended in the same proportion as those
effects are produced ; and hence arises the impossibility of
obtaining by their agency a perpetual effect ; or, in other words,
a perpetual motion. But the electro-motive force ascribed by
Yolta to the metals when in contact is a force which, as long
as a free course is allowed to the electricity it sets in motion,
is never expended, and continues to be exerted with undi-
minished power, in the production of a never-ceasing effect.
Against the truth of such a supposition the probabilities are
all but infinite."
This principle, which is little less than the doctrine of
conservation of energy applied to a voltaic cell, was reasserted
by Faraday. The process imagined by the contact school
" would," he wrote, "indeed be a creation of 'power -, like no other
force in nature." In all known cases energy is not generated,
but only transformed. There is no such thing in the world as
"a pure creation of force; a production of power without a
corresponding exhaustion of something to supply it."f
As time went on, each of the rival theories of the cell
became modified in the direction of the other. The contact
party admitted the importance of the surfaces at which the
metals are in contact with the liquid, where of course the chief
chemical action takes place ; and the chemical party confessed
their inability to explain the state of tension which subsists
before the circuit is closed, without introducing hypotheses just
as uncertain as that of contact force.
*Roget's Galvanism (1832), § 113.
•t Exp.Res., § 2071 (1840).
204 Faraday.
Faraday's own view on this point* was that a plate of
amalgamated zinc, when placed in dilute sulphuric acid, " has
power so far to act, by its attraction for the oxygen of the
particles'in contact with it, as to place the similar forces already
active between these and the other particles of oxygen and
the particles of hydrogen in the water, in a peculiar state of
tension or polarity, and probably also at the same time to
throw those of its own particles which are in contact with the
water into a similar but opposed state. Whilst this state is
retained, no further change occurs: but when it is relieved
by completion of the circuit, in which case the forces determined
in opposite directions, with respect to the zinc and the electro-
lyte, are found exactly competent to neutralize each other, then
a series of decompositions and recompositions takes place
amongst the particles of oxygen and hydrogen which constitute
the water, between the place of contact with the platina and
the place where the zinc is active : these intervening particles
being evidently in close dependence upon and relation to each
other. The zinc forms a direct compound with those particles
of oxygen which were, previously, in divided relation to both
it and the hydrogen : the oxide is removed by the acid, and a
fresh surface of zinc is presented to the water, to renew and
repeat the action."
These ideas were developed further by the later adherents
of the chemical theory, especially by Faraday's friend Christian
Friedrich Schonbein,f of Basle (6. 1799, d. 1868), the discoverer
of ozone. Schonbein made the hypothesis more definite by
assuming that when the circuit is open, the molecules of water
adjacent to the zinc plate are electrically polarized, the oxygen
side of each molecule being turned towards the zinc and being
negatively charged, while the hydrogen side is turned away
from the zinc and is positively charged. In the third quarter
* Exp. &»., § 949.
t Ann. d. Phys., Ixxviii (1849), p. 289, translated Archives des sc. phys., xiii
(1850), p. 192. Faraday and Schonbein for many years carried on a correspondence,
which has been edited by G. W. A. Kahlbaum and F. V. Darbishire : London,
Williams and Norgate.
Faraday. 205
of the nineteenth century, the general opinion was in favour
of some such conception as this. Helmholtz* attempted to-
grasp the molecular processes more intimately by assuming
that the different chemical elements have different attractive-
powers (exerted only at small distances) for the vitreous and
resinous electricities : thus potassium and zinc have strong
attractions for positive charges, while oxygen, chlorine, and
bromine have strong attractions for negative electricity. This
differs from Volta's original hypothesis in little else but
in assuming two electric fluids where Volta assumed only
one. It is evident that the contact difference of potential;
between two metals may be at once explained by Helmholtz's,
hypothesis, as it was by Volta's ; and the activity of the voltaic
cell may be referred to the same principles : for the two ions
of which the liquid molecules are composed will also possess
different attractive powers for the electricities, and may be
supposed to be united respectively with vitreous and resinous,
charges. Thus when two metals are immersed in the liquid,^
the circuit being open, the positive ions are attracted to the
negative metal and the negative ions to the positive metal,,
thereby causing a polarized arrangement of the liquid molecules
near the metals. When the circuit is closed, the positively
charged surface of the positive metal is dissolved into the fluid;,
and as the atoms carry their charge with them, the positive
charge on the immersed surface of this metal must be per-
petually renewed by a current flowing in the outer circuit.
It will be seen that Helmholtz did not adhere to Davy'ss
doctrine of the electrical nature of chemical affinity quite as,
simply or closely as Faraday, who preferred it in its most direct
and uncompromising form. " All the facts show us," he wrote,f
"that that power commonly called chemical affinity can be>
communicated to a distance through the metals and certain
forms of carbon ; that the electric current is only another form
of the forces of chemical affinity ; that its power is in proportion.
* In his celebrated memoir of 1847 on the Conservation, .o£.Huergy.
t Exp. Ties., § 918.
206 Faraday.
to the chemical affinities producing it ; that when it is deficient
in force it may be helped by calling in chemical aid, the want
in the former being made up by an equivalent of the latter;
that, in other words, the forces termed chemical affinity and
electricity are one and the same."
In the interval between Faraday's earlier and later papers
on the cell, some important results on the same subject were
published by Frederic Daniell (b. 1790, d. 1845), Professor of
Chemistry in King's College, London.* Daniell showed that
when a current is passed through a solution of a salt in water,
the ions which carry the current are those derived from the salt,
and not the oxygen and hydrogen ions derived from the water ;
this follows since a current divides itself between different mixed
electrolytes according to the difficulty of decomposing each, and
it is known that pure water can be electrolysed only with great
difficulty. Daniell further showed that the ions arising from
(say) sodium sulphate are not represented by Na20 and S03, but
by Na and S04 ; and that in such a case as this, sulphuric acid
is formed at the anode and soda at the cathode by secondary
action, giving rise to the observed evolution of oxygen and
hydrogen respectively at these terminals.
The researches of Faraday on the decomposition of chemical
compounds placed between electrodes maintained at different
potentials led him in 1837 to reflect on the behaviour of such
substances as oil of turpentine or sulphur, when placed in the
same situation. These bodies do not conduct electricity, and
are not decomposed ; but if the metallic faces of a condenser
are maintained at a definite potential difference, and if the
space between them is occupied by one of these insulating
substances, it is found that the charge on either face depends
on the nature of the insulating substance. If for any particular
insulator the charge has a value s times the value which it
would have if the intervening body were air, the number f
may be regarded as a measure of the influence which the
insulator exerts on the propagation of electrostatic action
* Phil. Trans., 1839, p. 97.
Faraday. 207
through it : it was called by Faraday the specific inductive
•capacity of the insulator.*
The discovery of this property of insulating substances or
dielectrics raised the question as to whether it could be
harmonized with the old ideas of electrostatic action. Consider,
for example, the force of attraction or repulsion between two
small electrically- charged bodies. So long as they are in air,
the force is proportional to the inverse square of the distance ;
but if the medium in which they are immersed be partly
changed — e.g., if a globe of sulphur be inserted in the intervening
space — this law is no longer valid : the change in the dielectric
affects the distribution of electric intensity throughout the
•entire field.
The problem could be satisfactorily solved only by forming
a physical conception of the action of dielectrics : and such a
conception Faraday now put forward.
The original idea had been promulgated long before by his
master Davy. Davy, it will be remembered,f in his explanation
of the voltaic pile, had supposed that at first, before chemical
decompositions take place, the liquid plays a part analogous to
that of the glass in a Leyden jar, and that in this is involved an
electric polarization of the liquid molecules.^ This hypothesis
was now developed by Faraday. Keferring first to his own work
on electrolysis, he asserted§ that the behaviour of a dielectric is
exactly the same as that of an electrolyte, up to the point at
which the electrolyte breaks down under the electric stress ; a
dielectric being, in fact, a body which is capable of sustaining
the stress without suffering decomposition.
" For," he argued,|| " let the electrolyte be water, a plate of
ice being coated with platina foil on its two surfaces, and these
* Exp. Res., § 1252 (1837). Cavendish had discovered specific inductive capacity
long before, but his papers were still unpublished.
t Cf. p. 77.
\ This is expressly stated in Davy's Elements of Chemical Philosophy (1812),
Div. i, § 7, where he lays it down that an essential " property of non-conductors"
is "to receive electrical polarities."
$ Exp. Res., §§ 1164, 1338, 1343, 1621.
|| Exp. Res., § 1164.
208 Faraday.
coatings connected with any continued source of the two
electrical powers, the ice will charge like a Leyden arrangement,
presenting a case of common induction, but no current will pass.
If the ice be liquefied, the induction will now fall to a certain
degree, because a current can now pass ; but its passing is
dependent upon a peculiar molecular arrangement of the particles
consistent with the transfer of the elements of the electrolyte in
opposite directions . . . As, therefore, in the electrolytic action,
induction appeared to bethejfe£ step,and decomposition the second
(the power of separating these steps from each other by giving
the solid or fluid condition to the electrolyte being in our hands) ;:
as the induction was the same in its nature as that through air,
glass, wax, &c., produced by any of the ordinary means ; and as
the whole effect in the electrolyte appeared to be an action of
the particles thrown into a peculiar or polarized state, I was
glad to suspect that common induction itself was in all cases an
action of contiguous particles, and that electrical action at a
distance (i.e., ordinary inductive action) never occurred except
through the influence of the intervening matter."
Thus at the root of Faraday's conception of electrostatic
induction lay this idea that the whole of the insulating medium
through which the action takes place is in a state of polarization
similar to that which precedes decomposition in an electrolyte.
" Insulators," he wrote,* " may be said to be bodies whose
particles can retain the polarized state, whilst conductors are
those whose particles cannot be permanently polarized."
The conception which he at this time entertained of the
polarization may be reconstructed from what he had already
written concerning electrolytes. He supposedf that in the
ordinary or unpolarized condition of a body, the molecules con-
sist of atoms which are bound to each other by the forces of
chemical affinity, these forces being really electrical in their
nature ; and that the same forces are exerted, though to a less
* Exp.Res., § 1338.
t This must not be taken to be more than an idea which Faraday mentioned as
present to his mind. He declined as yet to formulate a definite hypothesis.
Faraday. 209
degree, between atoms which belong to different molecules,
thus producing the phenomena of cohesion. When an electric
field is set up, a change takes place in the distribution of these
forces ; some are strengthened and some are weakened, the
effect being symmetrical about the direction of the applied
electric force.
Such a polarized condition acquired by a dielectric when
placed in an electric field presents an evident analogy to the
condition of magnetic polarization which is acquired by a mass
of soft iron when placed in a magnetic field ; and it was there-
fore natural that in discussing the matter Faraday should
introduce lines of electric force, similar to the lines of magnetic
force which he had employed so successfully in his previous
researches. A line of electric force he defined to be a curve
whose tangent at every point has the same direction as the
electric intensity.
The changes which take place in an electric field when the
dielectric is varied may be very simply described in terms of
lines of force. Thus if a mass of sulphur, or other substance of
high specific inductive capacity, is introduced into the field,
the effect is as if the lines of force tend to crowd into it : as
W. Thomson (Kelvin) showed later, they are altered in the
same way as the lines of flow of heat, in a case of steady con-
duction of heat, would be altered by introducing a body of
greater conducting power for heat. By studying the figures of
the lines of force in a great number of individual cases, Faraday
was led to notice that they always dispose themselves as if they
were subject to a mutual repulsion, or as if the tubes of force
had an inherent tendency to dilate.*
It is interesting to interpret by aid of these conceptions the
law of Priestley and Coulomb regarding the attraction between
two oppositely-charged spheres. In Faraday's view, the medium
intervening between the spheres is the seat of a system of
stresses, which may be represented by an attraction or tension
along the lines of electric force at every point, together with a
* Exp. Res., §§ 1224, 1297 (1837).
P
210 Faraday.
mutual repulsion of these lines, or pressure laterally. Where a
line of force ends on one of the spheres, its tension is exercised
on the sphere: in this way, every surface-element of each
sphere is pulled outwards. If the spheres were entirely
removed from each other's influence, the state of stress would be
uniform round each sphere, and the pulls on its surf ace -elements
would balance, giving no resultant force on the sphere. But
when the two spheres are brought into each other's presence,
the unit lines of force become somewhat more crowded together
on the sides of the spheres which face than on the remote sides,
and thus the resultant pull on either sphere tends to draw it
toward the other. When the spheres are at distances great
compared with their radii, the attraction is nearly proportional
to the inverse square of the distance, which is Priestley's law.
In the following year (1838) Faraday amplified* his theory
of electrostatic induction, by making further use of the analogy
with the induction of magnetism. Fourteen years previously
Poisson had imaginedf an admirable model of the molecular
processes which accompany magnetization; and this was now
applied with very little change by Faraday to the case of induc-
tion in dielectrics. " The particles of an insulating dielectric,"
he suggested, J " whilst under induction may be compared to a
series of small magnetic needles, or, more correctly still, to a
series of small insulated conductors. If the space round a
charged globe were filled with a mixture of an insulating
dielectric, as oil of turpentine or air, and small globular
conductors, as shot, the latter being at a little distance from
each other so as to be insulated, then these would in their
condition and action exactly resemble what I consider to be
the condition and action of the particles of the insulating
dielectric itself. If the globe were charged, these little con-
ductors would all be polar ; if the globe were discharged, they
would all return to their normal state, to be polarized again
upon the recharging of the globe/'
That this explanation accounts for the phenomena of specific
* Exp. Res., Series xiv. t Cf. p. 65. J Exp. Res., § 1679.
Faraday. 211
inductive capacity may be seen by what follows, which is
substantially a translation into electrostatical language of
Poisson's theory of induced magnetism.*
Let p denote volume-density of electric charge. For each
of Faraday's " small shot " the integral
JJJ pdx dy dz,
integrated throughout the shot, will vanish, since the total
charge of the shot is zero : but if r denote the vector (x, y, z),
the integral
J/J p r dx dy dz
will not be zero, since it represents the electric polarization of
the shot : if there are N shot per unit volume, the quantity
P = &!!! P r dx dy dz
will represent the total polarization per unit volume. If d
denote the electric force, and E the average value of d, P will
be proportional to E, say
P - (€ - 1) E.
By integration by parts, assuming all the quantities concerned
to vary continuously and to vanish at infinity, we have
+ p* D * (x> y> z} ** dy ds = "Iff* ^ p ** dy dz>
where ^ denotes an arbitrary function, and the volume-integrals
are taken throughout infinite space. This equation shows that
the polar-distribution of electric charge on the shot is equivalent
to a volume- distribution throughout space, of density
P = - div P.
Now the fundamental equation of electrostatics may in
suitable units be written,
div d = p ;
* W. Thomson (Kelvin), Camb. and Dub. Matb. Journal, November, 1845 ;
"W. Thomson's Papers on Electrostatics and Magnetism, § 43 sqq. ; F. 0. Mossotti,
Arcb. des sc. phys. (Geneva) vi (1847), p. 193 ; Mem. della Soc. Ital. Modena,
(2)xiv(1850), p. 49.
P 2
212 Faraday.
and this gives on averaging
div E = pi + jo,
where pi denotes the volume-density of free electric charge,
i.e. excluding that in the doublets ; or
div (E + P) = Plt
or div (* E) = plt
This is the fundamental equation of electrostatics, as modified
in order to take into account the effect of the specific inductive
capacity «.
The conception of action propagated step by step through a
medium by the influence of contiguous particles had a firm hold
on Faraday's mind, and was applied by him in almost every
part of physics. " It appears to me possible," he wrote in
1838,* " and even probable, that magnetic action may be
communicated to a distance by the action of the intervening
particles, in a manner having a relation to the way in which
the inductive forces of static electricity are transferred to a
distance ; the intervening particles assuming for the time more
or less of a peculiar condition, which (though with a very
imperfect idea) I have several times expressed by the term
electro-tonic state."^
The same set of ideas sufficed to explain electric currents.
Conduction, Faraday suggested,* might be " an action of
contiguous particles, dependent on the forces developed in
electrical excitement ; these forces bring the particles into a
state of tension or polarity ;§ and being in this state the
contiguous particles have a power or capability of communicating
these forces, one to the other, by which they are lowered and
discharge occurs."
* Exp Res., § 1729.
f This name had been devised in 1831 to express the state of matter subject to
magneto-electric induction ; cf. Exp.^Res., § 60.
J Exp. Res. iii, p. 513.
§ As in electrostatic induction in dielectrics.
Faraday. 213
After working strenuously for the ten years which followed
the discovery of induced currents, Faraday found in 1841 that
his health was affected ; and for four years he rested. A second
period of brilliant discoveries began in 1845.
Many experiments had been made at different times by
various investigators* with the purpose of discovering a
connexion between magnetism and light. These had generally
taken the form of attempts to magnetize bodies by exposure
in particular ways to particular kinds of radiation ; and a
successful issue had been more than once reported, only to be
negatived on re-examination.
The true path was first indicated by Sir John Herschel.
After his discovery of the connexion between the outward form
of quartz crystals and their property of rotating the plane of
polarization of light, Herschel remarked that a rectilinear
electric current, deflecting a needle to right and left all round
it, possesses a helicoidal dissymmetry similar to that displayed
by the crystals. " Therefore," he wrote,f " induction led me to
conclude that a similar connexion exists, and must turn up
somehow or other, between the electric current and polarized
light, and that the plane of polarization would be deflected by
magneto-electricity."
The nature of this connexion was discovered by Faraday,
who so far back as 1834J had transmitted polarized light
through an electrolytic solution during the passage of the
current, in the hope of observing a change of polarization.
This early attempt failed ; but in September, 1845, he varied
the experiment by placing a piece of heavy glass between the
poles of an excited electro-magnet ; and found that the plane
of polarization of a beam of light was rotated when the beam
travelled through the glass parallel to the lines of force of the
magnetic field. §
*e.g. by Morichini, of Rome, in 1813, Quart. Journ. Sci. xix, p. 338; by
Samuel Hunter Christie, of Cambridge, in 1825, Phil. Trans., 1826, p. 219 ; and
by Mary Somerville in the same year, Phil. Trans., 1826, p. 132.
t Sir. J. Herschel in Bence Jones's Life of Faraday, p. 205.
lExp. Res., § 951. \Ib., § 2152.
214 Faraday.
In the year following Faraday's discovery, Airy* suggested
a way of representing the effect analytically; as might have been
expected, this was by modifying the equations which had been
already introduced by MacCullagh for the case of naturally
active bodies. In Mac Cullagh's equations
|8^F=c2822r+ VZ
d^Y
the terms 83^/8#3 and 83F/8#3 change sign with x, so that the
rotation of the plane of polarization is always right-handed or
always left-handed with respect to the direction of the beam.
This is the case in naturally-active bodies ; but the rotation due
to a magnetic field is in the same absolute direction whichever
way the light is travelling, so that the derivations with respect
to x must be of even order. Airy proposed the equations
/82F 2 82F a£
I *-, * o 1 <*N 9 r^ *"\ /
dx* dt'
where p denotes a constant, proportional to the strength of the
magnetic field which is used to produce the effect. He remarked,
however, that instead of taking p dZ/dt and fj. 8 Y/dt as the additional
terms, it would be possible to take « 83^/8£3 and /u 83 F/8^3, or
im()3Z/dz?dt> and ju83F/8^28^, or any other derivates in which the
number of differentiations is odd with respect to t and even with
respect to x. It may, in fact, be shown by the method pre-
viously applied to Mac Cullagh's formulae that, if the equations
are
, 82F 8r+s^
8r+sF
where (r + s) is an odd number, the angle through which the
* Phil. Mag. xxviii (1846) p. 469.
Faraday. 215
plane of polarization rotates in unit length of path is a numerical
multiple of
where T denotes the period of the light. Now it was shown by
Verdet* that the magnetic rotation is approximately proportional
to the inverse square of the wave-length ; and hence we must
have
r + s= 3;
so that the only equations capable of correctly representing
Faraday's effect are either
w ^dx-dt
or
dt* W * dt
d'Z _ 2 d'Z
W '" °l ~W~^'W
The former pair arise, as will appear later, in Maxwell's
theory of rotatory polarization : the latter pair, which were
suggested in 1868 by Boussinesq,f follow from that physical
theory of the phenomenon which is generally accepted at the
present time.*
Airy's work on the magnetic rotation of light was limited
in the same way as MacCullagh's work on the rotatory power
of quartz ; it furnished only an analytical representation of the
effect, without attempting to justify the equations. The earliest
endeavour to provide a physical theory seems to have been
made in 1858, in the inaugural dissertation of Carl Neumann,
* Comptes Rendus, Ivi (1863), p. 630.
t Journal de Math., xiii (1868), p. 430.
J Fand Z being interpreted as components of electric force.
216 Faraday.
of Halle.* Neumann assumed that every element of an electric
current exerts force on the particles of the aether ; and in parti-
cular that this is true of the molecular currents which constitute
magnetization, although in this case the force vanishes except
when the aethereal particle is already in motion. If e denote the
displacement of the aethereal particle ra, the force in question
may be represented by the term
km [ e. K]
where K denotes the imposed magnetic field, and k denotes a
magneto-optic constant characteristic of the body. When this
term is introduced into the equations of motion of the aether,
they take the form which had been suggested by Airy ; whence
Neumann's hypothesis is seen to lead to the incorrect conclusion
that the rotation is independent of the wave-length.
The rotation of plane-polarized light depends, as Fresnel had
shown,f on a difference between the velocities of propagation of
the right-handed and left-handed circularly polarized waves into
which plane-polarized light may be resolved. In the case of
magnetic rotation, this difference was shown by Verdet to be
proportional to the component of the magnetic force in the
direction of propagation of the light; and CornuJ showed further
that the mean of the velocities of the right-handed and left-
handed waves is equal to the velocity of light in the medium
when there is no magnetic field. From these data, by Fresnel's
geometrical method, the wave- surf ace in the medium may be
obtained; it is found to consist of two spheres (one relating
to the right-handed and one to the left-handed light), each
identical with the spherical wave-surface of the unmagnetized
medium, displaced from each other along the lines of magnetic
force.§
The discovery of the connexion between magnetism and
* Explicare tentatur, quomodojiat, ut lucis planum polarisationis per vires el. vel
mag. declinetur. Halis Saxonum, 1858. The results were republished in a tract
Die magnetische Drehung der Polarisationsebene des Lichtes. Halle, 1863.
t Cf. p. 174. J Comptes Rendus, xcii (1881), p. 1368.
§ Cornu, Comptes Rendus, xcix (1884), p. 1045.
Faraday. 217
light gave interest to a short paper of a speculative character
which Faraday published* in 1846, under the title " Thoughts
on Kay- Vibrations." In this it is possible to trace the progress
of Faraday's thought towards something like an electro-magnetic
theory of light.
Considering first the nature of ponderable matter, he suggests
that an ultimate atom may be nothing else than a field of
force — electric, magnetic, and gravitational — surrounding a point-
centre ; on this view, which is substantially that of Michell and
Boscovich, an atom would have no definite size, but ought
rather to be conceived of as completely penetrable, and extend-
ing throughout all space ; and the molecule of a chemical
compound would consist not of atoms side by side, but of
" spheres of power mutually penetrated, and the centres even
coinciding."t
All space being thus permeated by lines of force, Faraday
suggested that light and radiant heat might be transverse
vibrations propagated along these lines of force. In this way
he proposed to " dismiss the aether," or rather to replace it by
lines of force between centres, the centres together with their
lines of force constituting the particles of material substances.
If the existence of a luminiferous aether were to be admitted,
Faraday suggested that it might be the vehicle of magnetic
force ; " for," he wrote in 1851,{ "it is not at all unlikely that
if there be an aether, it should have other uses than simply
the conveyance of radiations." This sentence may be regarded
as the origin of the electro-magnetic theory of light.
At the time when the " Thoughts on Eay -Vibrations " were
published, Faraday was evidently trying to comprehend every-
thing in terms of lines of force ; his confidence in which had
been recently justified by another discovery. A few weeks
after the first observation of the magnetic rotation of light, he
noticed § that a bar of the heavy glass which had been used in
* Phil. Mag. (3), xxviii (1846) : Exp. Res., iii, p. 447.
t Cf. Bence Jones's Life of Faraday, ii, p. 178.
J Exp. Res., $ 3075. § Phil. Trans., 1846, p. 21 : Exp. Res., § 2253.
218 Faraday.
this investigation, when suspended between the poles of an
electro-magnet, set itself across the line joining the poles : thus
behaving in the contrary way to a bar of an ordinary magnetic
substance, which would tend to set itself along this line. A
simpler manifestation of the effect was obtained when a cube
or sphere of the substance was used ; in such forms it showed
a disposition to move from the stronger to the weaker places
of the magnetic field. The pointing of the bar was then seen
to be merely the resultant of the tendencies of each of its
particles to move outwards into the positions of weakest
magnetic action.
Many other bodies besides heavy glass were found to
display the same property ; in particular, bismuth.* The name
diamagnetic was given to them.
" Theoretically," remarked Faraday, " an explanation of the
movements of the diamagnetic bodies might be offered in the
supposition that magnetic induction caused in them a contrary
state to that which it produced in magnetic matter ; i.e. that if
a particle of each kind of matter were placed in the magnetic
field, both would become magnetic, and each would have its
axis parallel to the resultant of magnetic force passing through
it ; but the particle of magnetic matter would have its north
and south poles opposite, or facing toward the contrary poles
of the inducing magnet, whereas with the diamagnetic particles
the reverse would be the case ; and hence would result approxi-
mation in the one substance, recession in the other. Upon
Ampere's theory, this view would be equivalent to the sup-
position that, as currents are induced in iron and magnetics
parallel to those existing in the inducing magnet or battery
wire, so in bismuth, heavy glass, and diamagnetic bodies, the
currents induced are in the contrary direction."
This explanation became generally known as the " hypothesis
of diamagnetic polarity " ; it represents diamagnetism as similar
* The repulsion of bismuth in the magnetic field had been previously observed
by A. Brugmans in 1778; Antonii Brugmans Magnetismus, Lugd. Bat., 1778.
t Exp. Res., § 2429.
Faraday. 219
to ordinary induced magnetism in all respects, except that the
direction of the induced polarity is reversed. It was accepted
by other investigators, notably by W. Weber, Pliicker, Eeich,
and Tyndall ; but was afterwards displaced from the favour of
its inventor by another conception, more agreeable to his peculiar
views on the nature of the magnetic field. In this second
hypothesis, Faraday supposed an ordinary magnetic or para-
magnetic* body to be one which offers a specially easy passage
to lines of magnetic force, so that they tend to crowd into
it in preference to other bodies ; while he supposed a dia-
magnetic body to have a low degree of conducting power for
the lines of force, so that they tend to avoid it. " If, then," he
reasoned,f "a medium having a certain conducting power occupy
the magnetic field, and then a portion of another medium or
substance be placed in the field having a greater conducting
power, the latter will tend to draw up towards the place of
greatest force, displacing the former." There is an electrostatic
effect to which this is quite analogous ; a charged body attracts
a body whose specific inductive capacity is greater than that of
the surrounding medium, and repels a body whose specific
inductive capacity is less; in either case the tendency is to
afford the path of best conductance to the lines of force .J
For some time the advocates of the "polarity" and
" conduction " theories of diarnagnetism carried on a contro-
versy which, indeed, like the controversy between the adherents
of the one-fluid and two-fluid theories of electricity, persisted
after it had been shown that the rival hypotheses were mathe-
matically equivalent, and that no experiment could be suggested
which would distinguish between them.
Meanwhile new properties of magnetizable bodies were being
discovered. In 1847 Julius Pliicker (b. 1801, d. 1868), Professor
of Natural Philosophy in the University of Bonn, while
repeating and extending Faraday's magnetic experiments,
* This term was introduced by Faraday, Exp. Res., § 2790.
t Exp. £es., § 2798.
J The mathematical theory of the motion of a magnetizable body in a non-
uniform field of force was discussed by "W. Thomson (Kelvin) in 1847.
220 Faraday.
observed* that certain uniaxal crystals, when placed between
the two poles of a magnet, tend to set themselves so that the
optic axis has the equatorial position. At this time Faraday
was continuing his researches ; and, while investigating the
diamagnetic properties of bismuth, was frequently embarrassed
by the occurrence of anomalous results. In 1848 he ascertained
that these were in some way connected with the crystalline
form of the substance, and showedf that when a crystal of
bismuth is placed in a field of uniform magnetic force (so that
no tendency to motion arises from its diamagnetism) it sets
itself so as to have one of its crystalline axes directed along
the lines of force.
At first he supposed this effect to be distinct from that
which had been discovered shortly before by Pliicker. " The
results," he wrote,J " are altogether very different from those
produced by diamagnetic action. They are equally distinct from
those discovered and described by Pliicker, in his beautiful
researches into the relation of the optic axis to magnetic action ;
for there the force is equatorial, whereas here it is axial. So
they appear to present to us a new force, or a new form of
force, in the molecules of matter, which, for convenience sake,
I will conventionally designate by a new word, as the magne-
crystallic force." Later in the same year, however, he recognized^
that " the phaenomena discovered by Pliicker and those of which
I have given an account have one common origin and cause."
The idea of the " conduction " of lines of magnetic force by
different substances, by which Faraday had so successfully
explained the phenomena of diamagnetism, he now applied to
the study of the magnetic behaviour of crystals. " If," he wrote,||
"the idea of conduction be applied to these magnecrystallic
bodies, it would seem to satisfy all that requires explanation in
their special results. A magnecrystallic substance would then
be one which in the crystallized state could conduct onwards, or
* Ann. d. Phys. Ixxii (1847), p. 315; Taylor's Scientific Memoirs, v, p. 353.
t Phil. Trans., 1849, p. 1 ; Exp. Res., § 2454.
i Exp. Res., § 2469. § Ibid., § 2605. || Ibid., § 2837.
Faraday. 221
permit the exertion of the magnetic force with more facility in
one direction than another ; and that direction would be the
magnecrystallic axis. Hence, when in the magnetic field, the
magnecrystallic axis would be urged into a position coincident
with the magnetic axis, by a force correspondent to that
difference, just as if two different bodies were taken, when the
one with the greater conducting power displaces that which is
weaker."
This hypothesis led Faraday to predict the existence of
another type of magnecrystallic effect, as yet unobserved. " If
such a view were correct/' he wrote,* " it would appear to
follow that a diamagnetic body like bismuth ought to be less
diamagnetic when its magnecrystallic axis is parallel to the
magnetic axis than when it is perpendicular to it. In the two
positions it should be equivalent to two substances having
different conducting powers for magnetism, and therefore if
submitted to the differential balance ought to present
differential phaenomena." This expectation was realized when
the matter was subjected to the test of experiment. f
The series of Faraday's " Experimental Researches in
Electricity " end in the year 1855. The closing period of his
life was quietly spent at Hampton Court, in a house placed at
his disposal by the kindness of the Queen ; and here on August
25th, 1867, he passed away.
Among experimental philosophers Faraday holds by uni-
versal consent the foremost place. The memoirs in which his
discoveries are enshrined will never cease to be read with
admiration and delight; and future generations will preserve
with an affection not less enduring the personal records and
familiar letters, which recall the memory of his humble and
unselfish spirit.
*Exp. Res., § 2839. ^ Ibid., § 2841.
222 The Mathematical Electricians of the
CHAPTEK VII.
THE MATHEMATICAL ELECTRICIANS OF THE MIDDLE OF THE
NINETEENTH CENTURY.
WHILE Faraday was engaged in discovering the laws of induced
currents in his own way, by use of the conception of lines of
force, his contemporary Franz Neumann was attacking the
same problem from a different point of view. Xeumann
preferred to take Ampere as his model ; and in 1845 published
a memoir,* in which the laws of induction of currents were
deduced by the help of Ampere's analysis.
Among the assumptions on which Neumann based his work
was a rule which had been formulated, not long after Faraday's
original discovery, by Emil Lenz,f and which may be enunciated
as follows : when a conducting circuit is moved in a magnetic
field, the induced current flows in such a direction that the
ponderomotive forces on it tend to oppose the motion.
Let ds denote an element of the circuit which is in motion,
and let C ds denote the component, taken in the direction of
motion, of the ponderomotive force exerted by the inducing
current on d$, when the latter is carrying unit current ; so that
the value of C is known from Ampere's theory. Then Lenz's
rule requires that the product of C into the strength of the
induced current should be negative. Xeumann assumed that
this is because it consists of a negative coefficient multiplying
the square of C\ that is, he assumed the induced electro-
motive force to be proportional to C. He further assumed it to
be proportional to the velocity v of the motion; and thus
obtained for the electromotive force induced in ds the expression
- ei-Cds,
where e denotes a constant coefficient. By aid of this formula,
•Berlin Abhandlungen, 1845, p. 1 ; 1848, p. 1 ; reprinted as Xo. 10 and
No. 36 of Ostwald's Klassiker-, translated Journal de Math, xiii (1848), p. 113.
t Ann. d. Phys. xxxi (1834), p. 483.
Middle of the Nineteenth Century. 223
in the earlier part* of the memoir, he calculated the induced
currents in various particular cases.
But having arrived at the formulae in this way, Neumann
noticedf a peculiarity in them which suggested a totally
different method of treating the subject. In fact, on examining
the expression for the current induced in a circuit which is in
motion in the field due to a magnet, it appeared that this
induced current depends only on the alteration caused by the
motion in the value of a certain function ; and, moreover, that
this function is no other than the potential of the ponderomotive
forces which, according to Ampere's theory, act between the
circuit, supposed traversed by unit current, and the magnet.
Accordingly, Neumann now proposed to reconstruct his
theory by taking this potential function as the foundation.
The nature of Neumann's potential, and its connexion
with Faraday's theory, will be understood from the following
considerations : —
The potential energy of a magnetic molecule M in a field
of magnetic intensity B is (B . M) ; and therefore the potential
energy of a current i flowing in a circuit s in this field is
where S denotes a diaphragm bounded by the circuit s ; as is
seen at once on replacing the circuit by its equivalent magnetic
shell S. If the field B be produced by a current i' flowing in a
circuit s', we have, by the formula of Biot and Savart,
1 *•'
curl —
* §§ 1-8. It may be remarked that Neumann, in making use of Ohm's law,
was (like everyone else at this time) unaware of the identity of electroscopic
force with electrostatic potential. t § 9.
224 The Mathematical Electricians of the
Hence, the mutual potential energy of the two currents is
- . dS
which hy Stokes's transformation may be written in the form
(ds.ds')
This expression represents the amount of mechanical work
which must be performed against the electro-dynamic pondero-
motive forces, in order to separate the two circuits to an infinite
distance apart, when the current-strengths are maintained
unaltered.
The above potential function has been obtained by con-
sidering the ponderomotive forces ; but it can now be connected
with Faraday's theory of induction of currents. For by
interpreting the expression
(B . dS)
If'
in terms of lines of force, we see that the potential function
represents the product of i into the number of unit-lines of
magnetic force due to s't which pass through the gap formed by
the circuit s ; and since by Faraday's law the currents induced
in s depend entirely on the variation in the number of these
lines, it is evident that the potential function supplies all that
is needed for the analytical treatment of the induced currents.
This was Neumann's discovery.
The electromotive force induced in a circuit s by the motion
of other circuits s', carrying currents i't is thus proportional to
the time-rate of variation of the potential
(ds.ds').
so that if we denote by a the vector
Middle of the Nineteenth Century. 225
which, of course, is a function of the position of the element ds
from which r is measured, then the electromotive force induced
in any circuit-element ds by any alteration in the currents
which give rise to a is
(a. ds).
The induction of currents is therefore governed by the vector a ;
this, which is generally known as the vector-potential, has from
Neumann's time onwards played a great part in electrical theory.
It may be readily interpreted in terms of Faraday's conceptions ;
for (a . ds) represents the total number of unit lines of magnetic
force which have passed across the line-element ds prior to the
instant t. The vector-potential may in fact be regarded as the
analytical measure of Faraday's electrotonic state*
While Neumann was endeavouring to comprehend the laws
of induced currents in an extended form of Ampere's theory,
another investigator was attempting a still more ambitious
project : no less than that of uniting electrodynamics into a
coherent whole with electrostatics.
Wilhelm Weber (6. 1804, d. 1890) was in the earlier part of
his scientific career a friend and colleague of Gauss at Gottingen.
In 1837, however, he became involved in political trouble. The
union of Hanover with the British Empire, which had subsisted
since the accession of the Hanoverian dynasty to the British
throne, was in that year dissolved by the operation of the Salic
law ; the Princess Victoria succeeded to the crown of England,
and her uncle Ernest- Augustus to that of Hanover. The new
king, who was a pronounced reactionary, revoked the free
constitution which the Hanoverians had for some time enjoyed ;
and Weber, who took a prominent part in opposing this action,
was deprived of his professorship. From 1843 to 1849, when
his principal theoretical researches in electricity were made,
he occupied a chair in the University of Leipzig.
The theory of Weber was in its origin closely connected
with the work of another Leipzig professor, Fechner, who in
1845f introduced certain assumptions regarding the nature of
* Cf. pp. 212, 272. t Ann. d. Phys. Lxiv (1845), p. 337.
Q
226 The Mathematical Electricians of the
electric currents. Fechner supposed every current to consist in
a streaming of electric charges, the vitreous charges travelling
in one direction, and the resinous charges, equal to them in
magnitude and number, travelling in the opposite direction with
equal velocity. He further supposed that like charges attract
each other when they are moving parallel to the same direction,
while unlike charges attract when they are moving in opposite
directions. On these assumptions he succeeded in bringing
Faraday's induction effects into connexion with Ampere's laws
of electrodynamics.
In 1846 Weber,* adopting the same assumptions as Fechner,
analysed the phenomena in the following way : —
The formula of Ampere for the ponderomotive force between
two elements ds, ds' of currents it i ', may be written
r ds ds r2 ds ds'
Suppose now that X units of vitreous electricity are contained
in unit length of the wire s, and are moving with velocity u ;
and that an equal quantity of resinous electricity is moving
with velocity u in the opposite direction ; so that
Let X', u', denote the corresponding quantities for the other
current; and let the suffix ! be taken to refer to the action
between the positive charges in the two wires, the suffix 2 to
the action between the positive charge in s and the negative
charge in s, the suffix 3 to the action between the negative
charge in s and the positive charge in s', and the suffix 4 to the
action between the negative charges in the two wires. Then
we have
'dr\ dr , dr
— = u — + u — ,,
dtji ds ds
* Elektrodynamische Maassbestimmungen, Leipzig Abhandl., 1846 : Ann. d.
Phys. hcxiii (1848), p. 193: English translation in Taylor's Scientific Memoirs,
v (1852), p. 489.
Middle of the Nineteenth Century. 227
and
fdzr\ zdzr , c?r <Fr
__ = u* __ + 2uu --—?-, + u 2 -j-7- •
df <fo* dsds ds*
By aid of these and the similar equations with the suffixes 3, 3, 4,
the equation for the ponderomotive force may be transformed
into the equation
d?r\ f dV
I nt
A A' tl&rtst' \ \ ///z /, \ ///* /„
F =
But this is the equation which we should have obtained
had we set out from the following assumptions : that the
ponderomotive force between two current-elements is the
resultant of the force between the positive charge in ds and the
positive charge in ds', of the force between the positive charge
in ds and the negative charge in dst etc. ; and that any two
electrified particles of charges e and e', whose distance apart
is r, repel each other with a force of magnitude
*l&
Two such charges would, of course, also exert on each other an
electrostatic repulsion, whose magnitude in these units would
be eec'/r2, where c denotes a constant* of the dimensions of a
velocity, whose value is approximately 3 x 1010 cm./sec. So
that on these assumptions the total repellent force would be
ee'cz f rr r*
«
* The units which have been adopted in the above investigation depend on the
electrodynamic actions of currents ; i.e. they are such that two unit currents flowing
in parallel circular circuits at a certain distance apart exert unit ponderomotive
force on each other. The quantity of electricity conveyed in unit time by such a
unit current is adopted as the unit~eharge. This unit charge is not identical with
the electrostatic unit charge, which is definedHqbe such that two unit charges at
unit distance apart repel each other with unit poniieiQmotive force. Hence the
necessity for introducing the factor c.
Q
228 The Mathematical Electricians of the
This expression for the force between two electric charges
was taken by Weber as the basis of his theory. Weber's is the
first of the electron-theories — a name given to any theory which
attributes the phenomena of electrodynamics to the agency
of moving electric charges, the forces on which depend not
only on the position of the charges (as in electrostatics), but
also on their velocity.
The latter feature of Weber's theory led its earliest critics
to deny that his law of force could be reconciled with the
principle of conservation of energy. They were, however,
mistaken on this point, as may be seen from the following
considerations. The above expression for the force between
two charges may be written in the form
where U denotes the expression
ee'c~
Consider now two material particles at distance r apart, whose
mechanical kinetic energy is T, and whose mechanical potential
energy is F, and which carry charges e and e'. The equations
of motion of these particles will be exactly the same as the
equations of motion of a dynamical system for which the
kinetic energy is
ee'i*
and the potential energy is
To such a system the principle of conservation of energy may be
applied : the equation of energy is, in fact,
m -rr 1 > • 6e ' G"
T + V - — ee r + - = constant.
2r r
Middle of the Nineteenth Century. 229
The first objection made to Weber's theory is thus disposed
of ; but another and more serious one now presents itself. The
occurrence of the negative sign with the term - ee'r^/Zr implies
that a charge behaves somewhat as if its mass were negative, so
that in certain circumstances its velocity might increase indefi-
nitely under the action of a force opposed to the motion. This
is one of the vulnerable points of Weber's theory, and has been
the object of much criticism. In fact,* suppose that one charged
particle of mass /z. is free to move, and that the other charges
are spread uniformly over the surface of a hollow spherical
insulator in which the particle is enclosed. The equation of
conservation of energy is
^(fi-ep)v*+ V= constant,
where e denotes the charge of the particle, v its velocity, V its
potential energy with respect to the mechanical forces which act
on it, and p denotes the quantity
- cos-(v.r)dS,
where the integration is taken over the sphere, and where o-
denotes the surface-density ; p is independent of the position of
the particle p within the sphere. If now the electric charge on
the sphere is so great that ep is greate^-tbsciTT^ then v2 and V
must increase and diminish together; which is evidently absurd.
Leaving this objection unanswered, we proceed to show how
Weber's law of force between electrons leads to the formulae
for the induction of currents.
The mutual energy of two moving charges is
~\ ~2cV'
°r ! * " L"v«r'Y' ~1'
r |_ *c r J
where v and v' denote the velocities of the charges ; so that the
* This example was given by Helmholtz, Journal fur Math. Ixxv (1873), p. 35 ;
Phil. Mag. xliv (1872), p. 530.
230 The Mathematical Electricians of the
mutual energy of two current-elements containing charges e, e
respectively of each kind of electricity, is
r3
If ds, ds' denote the lengths of the elements, and i, if the currents
in them, we have
ids = 2ev, i'ds' = 2«V ;
so the mutual energy of two current-elements is
nf
-(r.ds').(r.ds).
The mutual energy of ids with all the other currents is therefore
t(dt.a),
where a denotes a vector-potential
By reasoning similar to Neumann's, it may be shown that the
electromotive force induced in ds by any alteration in the rest
of the field is
-(ds.a);
and thus a complete theory of induced currents may be
constructed.
The necessity for induced currents may be inferred by
general reasoning from the first principles of Weber's theory.
When a circuit s moves in the field due to currents, the velocity
of the vitreous charges in s is, owing to the motion of s, not
equal and opposite to that of the resinous charges : this gives
rise to a difference in the forces acting on the vitreous and
resinous charges in s ; and hence the charges of opposite sign
separate from each other and move in opposite directions.
The assumption that positive and negative charges move
with equal and opposite velocities relative to the matter of
Middle of the Nineteenth Century. 231
the conductor is one to which, for various reasons which will
appear later, objection may be taken ; but it is an integral part
of Weber's theory, and cannot be excised from it. In fact,
if this condition were not satisfied, and if the law of force were
Weber's, electric currents would exert forces on electrostatic
charges at rest*; as may be seen by the following example.
Let a current flow in a closed circuit formed by arcs of two
concentric circles and the portions of the radii connecting their
extremities; then, if Weber's law were true, and if only one
kind of electricity were in motion, the current would evidently
exert an electrostatic force on a charge placed at the centre of
the circles. It has been shown,f indeed, that the assumption
of opposite electricities moving with equal and opposite veloci-
ties in a circuit is almost inevitable in any theory of the type
of Weber's, so long as the mutual action of two charges is
assumed to depend only on their relative (as opposed to their
absolute) motion.
The law of Weber is not the only one of its kind; an alterna-
tive to it was suggested by Bernhard Eiemann (b. 1826, d. 1866),
in a course of lectures which were delivered^ at Gottingen
in 1861, and which were published after his death by
K. Hattendorff. Kiemann proposed as the electrokinetic
energy of two electrons e (x, y, z) and e\xf, y\ z') the expression
this differs from the corresponding expression given by Weber
only in that the relative velocity of the two electrons is
substituted in place of the component of this velocity along
the radius vector. Eventually, as will be seen later, the laws
* This remark was first made by Clausius, Journal fur Math. Ixxxii (1877),
p. 86: the simple proof given above is due to Grassmann, Journal fur Math.
Ixxxiii (1877), p. 57.
t H. Lorberg, Journal fur Math. Ixxxiv (1878), p. 305.
J Schicere, Elektricitat und Magnetismus, nach den Vorlesungen von B. Riemann :
Hannover, 1875, p. 326. Another alternative to Weber's law had been discovered
by Gauss so far back as 1835, but was not published until after his death: cf.
Gauss' Werke, v, p. 616.
232 The Mathematical Electricians of the
of Riemann and Weber were both abandoned in favour of a
third alternative.
At the time, however, Weber's discovery was felt to be a
great advance ; and indeed it had, perhaps, the greatest share
in awakening mathematical physicists to a sense of the possi-
bilities latent in the theory of electricity. Beyond this, its
influence was felt in general dynamics ; for Weber's electro-
kinetic energy, which resembled kinetic energy in some respects
and potential energy in others, could not be precisely classified
under either head ; and its introduction, by helping to break
down the distinction which had hitherto subsisted between the
two parts of the kinetic potential, prepared the way for the
modern transformation-theory of dynamics.*
Another subject whose development was stimulated by the
work of Weber was the theory of gravitation. That gravitation
is propagated by the action of a medium, and consequently is a
process requiring time for its accomplishment, had been an article
of faith with many generations of physicists. Indeed, the
dependence of the force on the distance between the attracting
bodies seemed to suggest this idea ; for a propagation which is
truly instantaneous would, perhaps, be more naturally conceived
to be effected by some kind of rigid connexion between the
bodies, which would be more likely to give a force independent
of the mutual distance.
It is obvious that, if the simple law of Newton is abandoned,
there is a wide field of rival hypotheses from which to choose
its successor. The first notable attempt to discuss the question
was made by Laplace. f Laplace supposed gravity to be pro-
duced by the impulsion on the attracted body of a " gravific
fluid," which flows with a definite velocity toward the centre
of attraction — say, the sun. If the attracted body or planet
is in motion, the velocity of the fluid relative to it will be
compounded of the absolute velocity of the fluid and the
reversed velocity of the planet, and the force of gravity will
* Cf. "Whittaker, Analytical Dynamics, chapters ii, iii, xi.
t Meeanique Celeste, Livre x, chap, vii, § 22.
Middle of the Nineteenth Century. 233
act in the direction thus determined, its magnitude being
unaltered by the planet's motion. This amounts to supposing
that gravity is subject to an aberrational effect similar to that
observed in the case of light. It is easily seen that the modi-
fication thus introduced into Newton's law may be represented
by an additional perturbing force, directed along the tangent
to the orbit in the opposite sense to the motion, and pro-
portional to the planet's velocity and to the inverse square of
the distance from the sun. By considering the influence of
this force on the secular equation of the moon's motion, Laplace
found that the velocity of the gravific fluid must be at least a
hundred million times greater than that of light.
The assumptions made by Laplace are evidently in the
highest degree questionable; but the generation immediately
succeeding, overawed by his fame, seems to have found no way
of improving on them. Under the influence of Weber's ideas,
however, astronomers began to think of modifying Newton's
law by^ adding a term involving the velocities of the bodies.
Tisserand* in 1872 discussed the motion of the planets round
the sun on the supposition that the law of gravitation is the
same as Weber's law of electrodynamic action, so that the
force is
jp = «/_^r n . -?.r — J
* «.» ix nv^* i i«r^«i»
where / denotes the constant of gravitation, ra the mass of
the planet, // the mass of the sun, r the distance of the planet
from the sun, and h the velocity of propagation of gravitation.
The equations of motion may be rigorously integrated by
the aid of elliptic functions!; but the simplest procedure is
to write
* Comptes Rendus, Ixxv (1872), p. 760. Of. also Comptes Rendus, ex (1890),
p. 313, and Holzmiiller, Zeitschrif t f iir Math. u. Phys., 1870, p. 69.
t This had been done in an inaugural dissertation by Seegers, Gottingen, 1864.
234 The Mathematical Electricians of the
and, regarding F\ as a perturbing function, to find the variation
of the constants of elliptic motion. Tisserand showed that the
perturbations of all the elements are zero or periodic, and quite
insensible, except that of the longitude of perihelion, which has
a secular part. If A be assumed equal to the velocity of light,
the effect would be to rotate the major axis of the orbit of
Mercury in the direct sense 14" in a century.
Now, as it happened, a discordance between theory and
observation was known to exist in regard to the motion of
Mercury's perihelion ; for Le Verrier had found that the attrac-
tion of the planets might be expected to turn the perihelion
527" in the direct sense in a century, whereas the motion
actually observed was greater than this by 38". It is evident,
however, that only f of the excess is explained by Tisserand's
adoption of "Weber's law; and it seemed therefore that this
suggestion would prove as unprofitable as Le Terrier's own
hypothesis of an intra-mercurial planet. But it was found
later* that f of the excess could be explained by substituting
Eiemann's electrodynamic law for Weber's, and that a com-
bination of the laws of Biemann and Weber would give exactly
the amount desired.f
After the publication of his memoir on the law of force
between electrons, Weber turned his attention to the question
of diamagnetism, and developed Faraday's idea regarding the
explanation of diamagnetic phenomena by the effects of electric
currents induced in the diamagnetic bodies.^ Weber remarked
that if, with Ampere, we assume the existence of molecular
circuits in which there is no ohmic resistance, so that currents
can flow without dissipation of energy, it is quite natural to
suppose that currents would be induced in these molecular
* By Maurice Levy, Comptes Eendus, ex (1890), p. 545.
t The consequences of adopting the electrodynamic law of Clausius (for which
see later) were discussed by Oppenheim, Zur Frage nach der Fortpflanzungs-
geschwindigJceit der Gravitation, Wien, 1895.
I Leipzig Berichte, i (1847), p. 346 ; Ann. d. Phys. Ixxiii (1848), p. 241 ;
translated Taylor's Scientific Memoirs, v, p. 477 ; Abhandl. der K. Sachs. Ges. i
(1852), p. 483; Ann. d. Phys. Ixxxvii (1852), p. 145; trans. Tyndall and
Francis' Scientific Memoirs, p. 163.
Middle of the Nineteenth Century. 235
circuits if they were situated in a varying magnetic field ; and
he pointed out that such induced molecular currents would
confer upon the substance the properties characteristic of
dia magnetism.
The difficulty with this hypothesis is to avoid explaining too
much ; for, if it be accepted, the inference seems to be that all
bodies, without exception, should be diamagnetic. Weber escaped
from this conclusion by supposing that in iron and other
magnetic substances there exist permanent molecular currents,
which do not owe their origin to induction, and which, under
the influence of the impressed magnetic force, set themselves in
definite orientations. Since a magnetic field tends to give such
a direction to a pre-existing current that its course becomes
opposed to that of the current which would be induced by the
increase of the magnetic force, it follows that a substance stored
with such pre-existing currents would display the phenomena
of paramagnetism: t The bodies ordinarily called paramagnetic
are, according to this hypothesis, those bodies in which the
paramagnetism is strong enough to mask the diamagnetism.
The radical distinction which Weber postulated between the
natures of paramagnetism and diamagnetism accords with many
facts which have been discovered subsequently. Thus in 1895
P. Curie showed* that the magnetic susceptibility per gramme-
molecule is connected with the temperature by laws which are
different for paramagnetic and diamagnetic bodies. For the
former it varies in inverse proportion to the absolute tempe-
rature, whereas for diamagnetic bodies it is independent of the
temperature.
The conclusions which followed from the work of Faraday
and Weber were adverse to the hypothesis of magnetic fluids ;
for according to that hypothesis the induced polarity would be
in the same direction whether due to a change of orientation of
pre-existing molecular magnets, or to a fresh separation of
magnetic fluids in the molecules. " Through the discovery of
* Annales de Chimie (7) v (1845), p. 289.
236 The Mathematical Electricians of the
diamagnetism," wrote Weber* in 1852, "the hypothesis of
electric molecular currents in the interior of bodies is cor-
roborated, and the hypothesis of magnetic fluids in the interior
of bodies is refuted." The latter hypothesis is, moreover, unable
to account for the phenomena shown by bodies which are
strongly magnetic, like iron : for it is found that when the
magnetizing force is gradually increased to a very large value,
the magnetization induced in such bodies does not increase in
proportion, but tends to a saturation value This effect cannot
be explained on the assumptions of Poisson,but is easily deducible
from those of Weber; for, according to Weber's theory, the
magnetizing force merely orients existing magnets ; and when it
has attained such a value that all of them are oriented in the
same direction, there is nothing further to be done,
Weber's theory in its original form is, however, open to
some objection. If the elementary magnets are supposed to be
free to orient themselves without encountering any resistance,
it is evident that a very small magnetizing force would suffice
to turn them all parallel to each other, and thus would produce
immediately the greatest possible intensity of induced magnetism.
To overcome this difficulty, Weber assumed that every displace-
ment of a molecular circuit is resisted by a couple, which tends
to restore the circuit to its original orientation. This assump-
tion fails, however, to account for the fact that iron which
has been placed in a strong magnetic field does not return
to its original condition when it is removed from the field,
but retains a certain amount of residual magnetization.
Another alternative was to assume a frictional resistance
to the rotation of the magnetic molecules ; but if such a
resistance existed, it could be overcome only by a finite
magnetizing force ; and this inference is inconsistent with the
observation that some degree of magnetization is induced by
every force, however feeble.
The hypothesis which has ultimately gained acceptance is
that the orientation is resisted by couples which arise from the
* Ann. d. Phys. lxxxvii(1852), p. 145 ; Tyndall and Francis' Sci. Mem., p. 163.
Middle of the Nineteenth Century. 237
mutual action of the molecular magnets themselves. In the
unmagnetized condition the molecules " arrange themselves so
as to satisfy their mutual attraction by the shortest path, and
thus form a complete closed circuit of attraction," as D. E.
Hughes wrote* in 1883 ; when an external magnetizing force
is applied, these small circuits are broken up ; and at any stage
of the process a molecular magnet is in equilibrium under the
joint influence of the external force and the forces due to the
other molecules.
This hypothesis was suggested by Maxwell,t and has been
since developed by J. A. Ewing;J its consequences may be
illustrated by the following simple example§ : —
Consider two magnetic molecules, each of magnetic moment
m, whose centres are fixed at a distance c apart. When
undisturbed, they dispose themselves in the position of stable
equilibrium, in which they point in the same direction along
the line c. Now let an increasing magnetic force H be made
to act on them in a direction at right angles to the line c.
The magnets turn towards the direction of H ; and when
H attains the value Sm/c3, they become perpendicular to the
line c, after which they remain in this position, when H is
increased further. Thus they display the phenomena of induc-
tion initially proportional to the magnetizing force, and of
saturation. If the magnetizing force H be supposed to act
parallel to the line c, in the direction in which the axes
originally pointed, the magnets will remain at rest. But if H
acts in the opposite direction, the equilibrium will be stable
only so long as H is less than ra/c3 ; when H increases
beyond this limit, the equilibrium becomes unstable, and the
magnets turn over so as to point in the direction of H\ when
H is gradually decreased to zero, they remain in their new posi-
tions, thus illustrating the phenomenon of residual magnetism.
* Proc. Roy. Soc. xxxv (1883), p. 178.
f Treatise on Elect. $ May., § 443.
I Phil. Mag. xxx (1890), p. 205 ; Magnetic Induction in Iron atid other Metals,.
1891.
§ E. G. Gallop, Messenger of Math, xxvii (1897), p. 6.
238 The Mathematical Electricians of the
By taking a large number of such pairs of magnetic molecules,
originally oriented in all directions, and at such distances that
the pairs do not sensibly influence each other, we may
construct a model whose behaviour under the influence of
an external magnetic field will closely resemble the actual
behaviour of ferromagnetic bodies.
In order that the magnets in the model may come to rest
in their new positions after reversal, it will be necessary to
suppose that they experience some kind of dissipative force
which damps the oscillations ; to this would correspond in
actual magnetic substances the electric currents which would
be set up in the neighbouring mass when the molecular
magnets are suddenly reversed ; in either case, the sudden
reversals are attended by a transformation of magnetic energy
into heat.
The transformation of energy from one form to another is a
subject which was first treated in a general fashion shortly
before the middle of the nineteenth century. It had long been
known that the energy of motion and the energy of position
of a dynamical system are convertible into each other, and
that the amount of their sum remains invariable when the
system is self-contained. This principle of conservation of
dynamical energy had been extended to optics by Fresnel, who
had assumed* that the energy brought to an interface by
incident light is equal to the energy carried away from the
interface by the reflected and refracted beams. A similar
conception was involved in Eoget's and Faraday's defencef of
the chemical theory of the voltaic cell ; they argued that the
work done by the current in the outer circuit must be provided
at the expense of the chemical energy stored in the cell, and
showed that the quantity of electricity sent round the circuit
is proportional to the quantity of chemicals consumed, while
its tension is proportional to the strength of the chemical
affinities concerned in the reaction. This theory was extended
*Cf. p. 133. tCf. p. 203.
Middle of the Nineteenth Century. 239
and completed by James Prescott Joule, of Manchester, in 1841.
Joule, who believed* that heat is producible from mechanical
work and convertible into it, measuredf the amount of heat
evolved in unit time in a metallic wire, through which a
current of known strength was passed; he found the amount
to be proportional to the resistance of the wire multiplied by
the square of the current- strength ; or (as follows from Ohm's
law) to the current-strength multiplied by the difference of
electric tensions at the extremities of the wire.
The quantity of energy yielded up as heat in the outer
circuit being thus known, it became possible to consider the
transference of energy in the circuit as a whole. " When,"
wrote Joule, " any voltaic arrangement, whether simple or
compound, passes a current of electricity through any substance,
whether an electrolyte or not, the total voltaic heat which is
generated in any time is proportional to the number of atoms
which are electrolyzed in each cell of the circuit, multiplied
by the virtual intensity of the battery : if a decomposing cell
be in the circuit, the virtual intensity of the battery is reduced
in proportion to its resistance to electrolyzation." In the same
year hej enhanced the significance of this by showing that the
quantities of heat which are evolved by the combustion of the
equivalents of bodies are proportional to the intensities of their
affinities for oxygen, as measured by the electromotive force
of a battery required to decompose the oxide electrolytically.
The theory of Koget and Faraday, thus perfected by Joule,
enables us to trace quantitatively the transformations of energy
in the voltaic cell and circuit. The primary source of energy
is the chemical reaction : in a Daniell cell, ZnjZn SOJCu S04|Cu,
for instance, it is the substitution of zinc for copper as the
partner of the sulphion. The strength of the chemical affinities
concerned is in this case measured by the difference of the heats
of formation of zinc sulphate and copper sulphate ; and it is
*Cf. p. 33.
t Phil. Mag. xix (1841), p. 260 ; Joule's Scientific Papers i, p. 60.
I Phil. Mag. xx (1841), p. 98 : cf. also Phil. Mag. xxii (1843), p. 204.
240 The Mathematical Electricians of the
this which determines the electromotive force of the cell.*
The amount of energy which is changed from the chemical to
the electrical form in a given interval of time is measured by
the product of the strength of the chemical affinity into the
quantity of chemicals decomposed in that time, or (what is the
same thing) by the product of the electromotive force of the
cell into the quantity of electricity which is circulated. This
energy may be either dissipated as heat in conformity to
Joule's law, or otherwise utilized in the outer circuit.
The importance of these principles was emphasized by
Hermann von Helmholtz (b. 1821, d. 1894), in a memoir which
was published in 1847, and which will be more fully noticed
presently, and by W. Thomson (Lord Kelvin) in 1851f; the
equations have subsequently received only one important
modification, which is due to Helmholtz.:}: Helmholtz pointed
out that the electrical energy furnished by a voltaic cell need
not be derived exclusively from the energy of the chemical
reactions : for the cell may also operate by abstracting heat-
energy from neighbouring bodies, and converting this into
electrical energy. The extent to which this takes place is
determined by a law which was discovered in 1855 by Thomson. §
Thomson showed that if E denotes the " available energy," i.e.,
possible output of mechanical work, of a system maintained
at the absolute temperature T, then a fraction
TdE
fidT
of this work is obtained, not at the expense of the thermal or
* The heat of formation of a gramme-molecule of ZnS04 is greater than the heat
of formation of a gramme-molecule of CuSO* by about 50,000 calories ; and with
divalent metals, 46,000 calories per gramme- molecule corresponds to ane.m.f. of one
volt ; so the e.m.f. of a Daniell cell should be 50/46 volts, which is nearly the
case.
t Kelvin's Math, and Phys. Papers, i, pp. 472, 490.
J Berlin Sitzungsber., 1882, pp. 22, 825 ; 1883, p. 647.
§ Quart. Journ. Math., April, 1855 ; Kelvin's Math, and Phys. Papers, i,
p. 297, eqn. (7).
Middle of the Nineteenth Century. 241
chemical energy of the system itself, but at the expense of the
thermal energy of neighbouring bodies. Now in the case of
the voltaic cell, the principle of Eoget, Faraday, and Joule is
expressed by the equation
^ = A,
where E denotes the available or electrical energy, which is
measured by the electromotive force of the cell, and where X
denotes the heat of the chemical reaction which supplies this
energy. In accordance with Thomson's principle, we must
replace this equation by
F \ 4- TdE
^=X + TdT'
which is the correct relation between the electromotive force
of a cell and the energy of the chemical reactions which occur
in it. In general the term A is much larger than the term
T dEjdT ; but in certain classes of cells — e.g., concentration-
cells — A is zero; in which case the whole of the electrical
energy is procured at the expense of the thermal energy of
the cells' surroundings.
Helmholtz's memoir of 1847, to which reference has already
been made, bore the title, " On the Conservation of Force." It
was originally read to the Physical Society of Berlin*; but
though the younger physicists of the Society received it with
enthusiasm, the prejudices of the older generation prevented
its acceptance for the Annalen der Physik ; and it was eventually
published as a separate treatise.f
In this memoir it was asserted* that the conservation of
* On July 23rd, 1847.
t Berlin, G. A. Reimer. English Translation in Tyndall & Francis' Scientific
Memoirs, p. 114. The publisher, to Helmholtz's "great surprise," gave him an
honorarium. Cf. Hermann von Helmholtz, by Leo Koenigsbeiger ; English
translation by F. A. Welby.
j Helmholtz had been partly anticipated by "W. R. Grove, in his lectures on
the Correlation of Physical Forces, which were delivered in 1843 and published in
1846. Grove, after asserting that heat is " purely dynamical " in its nature, and
that the various " physical forces " may be transformed into each other, remarked :
" The great problem which remains to be solved, in regard to the correlation
of physical forces, is the establishment of their equivalent of power, or their
measurable relation to a given standard."
P.
242 The Mathematical Electricians of the
energy is a universal principle of nature : that the kinetic and
potential energy of dynamical systems may be converted into
heat according to definite quantitative laws, as taught by
Kumford, Joule, and Eobert Mayer* ; and that any of these
forms of energy may be converted into the chemical, electro-
static, voltaic, and magnetic forms. The latter Helmholtz
examined systematically.
Consider first the energy of an electrostatic field. It will
be convenient to suppose that the system has been formed by
continually bringing from a very great distance infinitesimal
quantities of electricity, proportional to the quantities already
present at the various points of the system ; so that the charge
is always distributed proportionally to the final distribution.
Let e typify the final charge at any point of space, and V the
final potential at this point. Then at any stage of the process
the charge and potential at this point will have the values \e
and A F, where A denotes a proper fraction. At this stage let
charges ed\ be brought from a great distance and added to the
charges \e. The work required for this is
so the total work required in order to bring the system from
infinite dispersion to its final state is
fi
or
By reasoning similar to that used in the case of electrostatic
distributions, it may be shown that the energy of a magnetic
field, which is due to permanent magnets and which also
contains bodies susceptible to magnetic induction, is
\
where p0 denotes the density of Poisson's equivalent magnetiza-
* Julius Robert Mayer (b. 1814, d. 1878), who was a medical man in Heilbronn,
asserted the equivalence of heat and work in 1842, Annal. d. Chemie, xlii, p. 233 ;
his memoir, like that of Helmholtz, was first declined by the editors of the
Annalen der Physik. An English translation of one of Mayer's memoirs was
printed in Phil. Mag. xxv (1863), p. 493.
Middle of the Nineteenth Century. 243
tion, for the permanent magnets only, and $ denotes the magnetic
potential.*
Helmholtz, moreover, applied the principle of energy to
systems containing electric currents. For instance, when a
magnet is moved in the vicinity of a current, the energy taken
from the battery may be equated to the sum of that expended
as Joulian heat, and that communicated to the magnet by the
electromagnetic force : and this equation shows that the current
is not proportional to the electromotive force of the battery,
i.e. it reveals the existence of Faraday's magneto-electric
induction. As, however, Helmholtz was at the time un-
acquainted with the conception of the electrokinetic energy
stored in connexion with a current, his equations were for the
most part defective. But in the case of the mutual action of
a current and a permanent magnet, he obtained the correct
result that the time-integral of the induced electromotive
force in the circuit is equal to the increase which takes
place in the potential of the magnet towards a current of a
certain strength in the circuit.
The correct theory of the energy of magnetic and electro-
magnetic fields is due mainly to W. Thomson (Lord Kelvin).
Thomson's researches on this subject commenced with one or
two short investigations regarding the ponderomotive forces
which act on temporary magnets. In 1847 he discussed t the
case of a small iron sphere placed in a magnetic field, showing
that it is acted on by a ponderomotive force represented by
- grad cR~, where c denotes a constant, and R denotes the magnetic
force of the field ; such a sphere must evidently tend to move
towards the places where E' is greatest. The same analysis
may be applied to explain why diamagnetic bodies tend to
move, as in Faraday's experiments, from the stronger to the
weaker parts of the field.
* We suppose all transitions to be continuous, so as to avoid the necessity for
writing surf ace -integrals separately.
tCamb. and Dub. Matb. Journal, ii (1847), p. 230; W. Thomson's Papers
on Electrostatics and Magnetism, p. 499; cf. also Phil. Mag. xxxvii (1850),
p. 241.
R 2
24:4 The Mathematical Electricians of the
Two years later Thomson presented to the Koyal Society a
memoir* in which the results of Poisson'a theory of magnetism
were derived from experimental data, without making use of
the hypothesis of magnetic fluids ; and this was followed in
1850 by a second memoir,f in which Thomson drew attention
to the fact previously noticed by Poisson,J that the magnetic
intensity at a point within a magnetized body depends on the
shape of the small cavity in which the exploring magnet is
placed. Thomson distinguished two vectors ;§ one of these, by
later writers generally denoted by B, represents the magnetic
intensity at a point situated in a small crevice in the
magnetized body, when the faces of the crevice are at right
angles to the direction of magnetization ; the vector B is always
circuital. The other vector, generally denoted by H, represents
the magnetic intensity in a narrow tubular cavity tangential
to the direction of magnetization ; it is an irrotational vector.
The magnetic potential tends at any point to a limit which is
independent of the shape of the cavity in which the point is
situated ; and the space-gradient of this limit is identical with
H. Thomson called B the " magnetic force according to the
electro-magnetic definition," and H the " magnetic force accord-
ing to the polar definition " ; but the names magnetic induction
and magnetic force, proposed by Maxwell, have been generally
used by later writers.
It may be remarked that the vector to which Faraday
applied the term " magnetic force," and which he represented
by lines of force, is not H, but B ; for the number of unit lines
of force passing through any gap must depend only on the gap,
and not on the particular diaphragm filling up the gap, across
which the flux is estimated ; and this can be the case only if the
vector which is represented by the lines of force is a circuital
vector.
* Phil. Trans., 1851, p. 243 ; Thomson's Papers on Elect, and Mag., p. 345.
t Phil. Trans., 1851, p. 269 ; Papers on Elect, and Nay., p. 382.
I Of. p. 64.
§ Loc. cit., § 78 of the original paper, and § 517 of the reprint^
Middle of the Nineteenth Century. 245
Thomson introduced a number of new terms into magnetic
science — as indeed he did into every science in which he was
interested. The ratio of the measure of the induced magnetiza-
tion I,-, in a temporary magnet, to the magnetizing force H,
he named the susceptibility ; it is positive for paramagnetic and
negative for diamagnetic bodies, and is connected with Poisson's
constant kp* by the relation
3 if
t\jp
= SFTv
where K denotes the susceptibility. By an easy extension of
Poisson's analysis it is seen that the magnetic induction and
magnetic force are connected by the equation
B = H + 47rl,
where I denotes the total intensity of magnetization : so if I0
denote the permanent magnetization, we have
B = H + 47rl» + 47rl,,,
= )uH + 47rI0,
where //, denotes (1 + 4™) : //, was called by Thomson the
permeability.
In 1851 Thomson extended his magnetic theory so as to
include magnecrystallic phenomena. The mathematical founda-
tions of the theory of magnecrystallic action had been laid by
anticipation, long before the experimental discovery of the
phenomenon, in a memoir read by Poisson to the Academy in
February, 1824. Poisson, as will be remembered, had supposed
temporary magnetism to be due to " magnetic fluids," movable
within the infinitely small " magnetic elements " of which he
assumed magnetizable matter to be constituted. He had not
overlooked the possibility that in crystals these magnetic
elements might be non-spherical (e.g. ellipsoidal), and symmetri-
cally arranged ; and had remarked that a portion of such
a crystal, when placed in a magnetic field, would act in a
manner depending on its orientation. The relations connecting
* Cf. p. 65.
246 The Mathematical Electricians of the
the induced magnetization I with the magnetizing force H he
had given in a form equivalent to
( Ix = aHx + b'ffy + c"ffz,
Iy = a"Hx + bHy + c'HZ)
Iz = a'Hr + b"Hy + cHz.
Thomson now* showed that the nine coefficients a, b' ', c" . . .,
introduced by Poisson, are not independent of each other. For
a sphere composed of the magnecrystalline substance, if placed
in a uniform field of force, would be acted on by a couple : and
the work done by this couple when the sphere, supposed of
unit volume, performs a complete revolution round the axis of x
may be easily shown to be 7rH(l - H^j IP) (- &" + c). But this
work must be zero, since the system is restored to its primitive
condition ; and hence ~b" and c must be equal. Similarly e" = a,
and a" = bf. By change of axes three more coefficients may be
removed, so that the equations may be brought to the form
777" T TT T TT
JC ~ Kl/Zx, Iy = K-lJily, 1Z = Ka/Zz,
where KI, KZ, K3 may be called the principal magnetic suscepti-
bilities.
In the same year (1851) Thomson investigated the energy
which, as was evident from Faraday's work on self-induction,
must be stored in connexion with every electric current. He
showed that, in his own words, f " the value of a current in a
closed conductor, left without electromotive force, is the
quantity of work that would be got by letting all the infinitely
small currents into which it may be divided along the lines of
motion of the electricity come together from an infinite distance,
and make it up. Each of these ' infinitely small currents ' is of
course in a circuit which is generally of finite length ; it is the
section of each partial conductor and the strength of the current
in it that must be infinitely small."
* Phil. Mag. (4) i (1851), p. 177: Papers on Electrostatics and Magnetism,
p. 471.
t Papers on Electrostatics and Magnetism, p. 446.
Middle of the Nineteenth Century. 247
Discussing next the mutual energy due to the approach of a
permanent magnet and a circuit carrying a current, he arrived
at the remarkable conclusion that in this case there is no
electrokinetic energy which depends on the mutual action ; the
energy is simply the sum of that due to the permanent magnets
and that due to the currents. If a permanent magnet is
caused to approach a circuit carrying a current, the electromotive
force acting in the circuit is thereby temporarily increased ; the
amount of energy dissipated as Joulian heat, and the speed of
the chemical reactions in the cells, are temporarily increased also.
But the increase in the Joulian heat is exactly equal to the
increase in the energy derived from consumption of chemicals,
together with the mechanical work done on the magnet by the
operator who moves it ; so that the balance of energy is perfect,
and none needs to be added to or taken from the electrokinetic
form. It will now be evident why it was that Helmholtz
escaped in this case the errors into which he was led in other
cases by his neglect of electrokinetic energy ; for in this case
there was no electrokinetic energy to neglect.
Two years later, in 1853, Thomson* gave a new form to the
expression for the energy of a system of permanent and
temporary magnets.
We have seen that the energy of such a system is represented
by
where p0 denotes the density of Poisson's equivalent magnetiza-
tion for the permanent magnets, and <f> denotes the magnetic
potential, and where the integration may be extended over the
whole of space. Substituting for pn its value - div I0,f the
expression may be written in the form
- J
<£ div Io dx dydz ;
*Proc. Glasgow Phil. Soc. iii (1853), p. 281; Kelvin's Math, and Phys.
Papers, i, p. 521. t Cf . p. fi4.
248 The Mathematical Electricians of the
or, integrating by parts,
(!«, . grad <£) dx dy dz, or - J (H . I0) dx dy dz.
Since B = yu,H + 47rI0, this expression may be written in the
form
-— (H.
offJJJ
but the former of these integrals is equivalent to
>fff
(B . grad <£) dx dydz, or - <£ div B dx dy dz,
which vanishes, since B is a circuital vector. The energy of the
field, therefore, reduces to
1
BIT,
integrated over all space; which is equivalent to Thomson's
form.*
In the same memoir Thomson returned to the question of the
energy which is possessed by a circuit in virtue of an electric
current circulating in it. As he remarked, the energy may
be determined by calculating the amount of work which
must be done in and on the circuit in order to double the
circuit on itself while the current is sustained in it with
constant strength; for Faraday's experiments show that a
circuit doubled on itself has no stored energy. Thomson found
that the amount of work required may be expressed in the form
\Li*, where i denotes the current strength, and L, which is
called the coefficient of self-induction^ depends only on the form of
the circuit.
It may be noticed that in the doubling process the inherent
* The form actually given by Thomson was
— fff (*E? — lA d-d
Sir}}} \:-.*)
which reduces to the above when we neglect that part of I2 which is due to the
permanent magnetism, over which we have no control.
Middle of the Nineteenth Century. 249
electrodynamic energy is being given up, and yet the operator is
doing positive work. The explanation of this apparent paradox
is that the energy derived from both these sources is being
used to save the energy which would otherwise be furnished by
the battery, and which is expended in Joulian heat.
Thomson next proceeded* to show that the energy which is
stored in connexion with a circuit in which a current is flowing
may be expressed as a volume-integral extended over the whole
of space, similar to the integral by which he had already
represented the energy of a system of permanent and temporary
magnets. The theorem, as originally stated by its author,
applied only to the case of a single circuit; but it may be
established for a system formed by any number of circuits in
the following way : —
If N8 denote the number of unit tubes of magnetic induction
which are linked with the &h circuit, in which a current is is
flowing, the electrokinetic energy of the system is JSJV,^; which
may be written |2/r, where /r denotes the total current flowing
through the gap formed by the rth unit tube of magnetic induc-
tion. But if H denote the (vector) magnetic force, and H its
numerical magnitude, it is known that (l/4?r) J Hds, integrated
along a closed line of magnetic induction, measures the total
current flowing through the gap formed by the line. The
energy is therefore (l/8?r)S jffds, the summation being extended
over all the unit tubes of magnetic induction, and the integra-
tion being taken along them. But if dS denote the cross-section
of one of these tubes, we have BdS = 1, where B denotes the
numerical magnitude of the magnetic induction B : so the energy
is (1 1 'Sir) SBdS / Hds ; and as the tubes fill all space, we may
replace 'S.dSjds by ^dxdydz. Thus the energy takes the form
(l/8?r) JJf BHdxdydz, where the integration is extended over the
whole of space ; and since in the present case B = pH, the energy
may also be represented by (Il8v)ffffjjrdxdydz.
* Nichols* Cyclopaedia, 2nd ed., 1860, article " Magnetism, dynamical
relations of; " reprinted in Thomson's Papers on Elect, and Mag., p. 447, and
his Math, and Phys. Papers, p. 532.
250 The Mathematical Electricians of the
But this is identical with the form which was obtained for
a field due to permanent and temporary magnets. It thus
appears that in all cases the stored energy of a system of
electric currents and permanent and temporary magnets is
-' dxdydz,
where the integration is extended over all space.
It must, however, be remembered that this represents only
what in thermodynamics is called the " available energy " ; and
it must further be remembered that part even of this available
energy may not be convertible into mechanical work within the
limitations of the system : e.g., the electrokinetic energy of a
current flowing in a single closed perfectly conducting circuit
cannot be converted into any other form so long as the circuit
is absolutely rigid. All that we can say is that the changes in
this stored electrokinetic energy correspond to the work furnished
by the system in any change.
The above form suggests that the energy may not be localized
in the substance of the circuits and magnets, but may be distri-
buted over the whole of space, an amount (pH2 /Sir) of energy
being contained in each unit volume. This conception was
afterwards adopted by Maxwell, in whose theory it is of
fundamental importance.
While Thomson was investigating the energy stored in
connexion with electric currents, the equations of flow of the
currents were being generalized by Gustav Kirchhoff (b. 1824,
d. 1887). In 1848 Kirchhoff* extended Ohm's theory of linear
conduction to the case of conduction in three dimensions ; this
could be done without much difficulty by making use of the
analogy with the flow of heat, which had proved so useful to
Ohm. In Kirchhoff s memoir a system is supposed to be
formed of three-dimensional conductors, through which steady
currents are flowing. At any point let V denote the " tension "
or " electroscopic force " — a quantity the significance of which
*Ann.d. Phys. Ixxv (1848), p. 189: Kirchhoff's Ges. AbhandL, p. 33.
Middle of tke Nineteenth Century. 425l
in electrostatics was not yet correctly known. Then, within
the substance of any homogeneous conductor, the function V
must satisfy Laplace's equation V- V= 0 ; while at the air-surface
of each conductor, the derivate of V taken along the normal
must vanish. At the interface between two conductors formed
of different materials, the function V has a discontinuity,
which is measured by the value of Volta's contact force for the
two conductors ; and, moreover, the condition that the current
shall be continuous across such an interface requires that
Jed VfoN shall be continuous, where k denotes the ohmic specific
conductivity of the conductor, and 3/3^ denotes differentiation
along the normal to the interface. The equations which have
now been mentioned suffice to determine the flow of electricity
in the system.
Kirchhoff also showed that the currents distribute them-
selves in the conductors in such a way as to generate the least
possible amount of Joulian heat ; as is easily seen, since the
quantity of Joulian heat generated in unit time is
where k, as before, denotes the specific conductivity ; and this
integral has a stationary value when V satisfies the equation
a /ar\ a
Kirchhoff next applied himself to establish harmony between
electrostatical conceptions and the theory of Ohm. That
theory had now been before the world for twenty years, and
had been verified by numerous experimental researches ; in
particular, a careful investigation was made at this time (1848)
by Kudolph Kohlrausch (b. 1809, d. 1858), who showed* that
the difference of the electric " tensions " at the extremities of a
voltaic cell, measured electrostatically with the circuit open,
was for different cells proportional to the electromotive force
*Ann. d. Phys. Ixxv (1848), p. 220.
252 The Mathematical Electricians of the
measured by the electrodynamic effects of the cell with the
circuit closed ; and, further,* that when the circuit was closed,
the difference of the tensions, measured electrostatically, at any
two points of the outer circuit was proportional to the ohmic
resistance existing between them. But in spite of all that had
been done, it was still uncertain how " tension," or " electro-
scopic force," or " electromotive force " should be interpreted
in the language of theoretical electrostatics ; it will be
remembered that Ohm himself, perpetuating a confusion which
had originated with Volta, had identified electroscopic force
with density of electric charge, and had assumed that the
electricity in a conductor is at rest when it is distributed
uniformly throughout the substance of the conductor.
The uncertainty was finally removed in 1849 by Kirchhoff,f
who identified Ohm's electroscopic force with the electrostatic
potential. That this identification is correct may be seen by
comparing the different expressions which have been obtained
for electric energy; Helmholtz's expression^ shows that the
energy of a unit charge at any place is proportional to the
value of the electrostatic potential at that place ; while Joule's
result§ shows that the energy liberated by a unit charge in
passing from one place in a circuit to another is proportional
to the difference of the electric tensions at the two places. It
follows that tension and potential are the same thing.
The work of Kirchhoff was followed by several other
investigations which belong to the borderland between electro-
statics and electrodynamics. One of the first of these was the
study of the Leyden jar discharge.
Early in the century Wollaston, in the course of his experi-
ments on the decomposition of water, had observed that when
the decomposition is effected by a discharge of static electricity,
the hydrogen and oxygen do not appear at separate electrodes ;
but that at each electrode there is evolved a mixture of the
* Ann. d. Phys, Ixxviii (1849), p. 1.
f Ib. Ixxviii (1849), p. 506 ; Kirchhoff's Get. Abhandl, p. 49 ; Phil. Mag. (3),
xxxvii (1850), p. 463.
I Cf. p. 242. § Cf. p. 239.
Middle of the Nineteenth Century. 253
gases, as if the current had passed through the water in both
directions. After this F. Savary* had noticed that the
discharge of a Ley den jar magnetizes needles in alternating
layers, and had conjectured that " the electric motion during
the discharge consists of a series of oscillations." A similar
remark was made in connexion with a similar observation by
Joseph Henry (ft. 1799, d. 1878), of Washington, in 1842.f
" The phenomena," he wrote, " require us to admit the existence
of a principal discharge in one direction, and then several reflex
actions backward and forward, each more feeble than the
preceding, until equilibrium is restored." Helmholtz had
repeated the same suggestion in his essay on the conservation
of energy : and in 1853 W. Thomson J verified it, by
investigating the mathematical theory of the discharge, as
follows : —
Let C denote the capacity of the jar, i.e., the measure of the
charge when there is unit difference of potential between "the
coatings ; let R denote the ohmic resistance of the discharging
circuit, and L its coefficient of self-induction. Then if at
any instant t the charge of the condenser be Q, and the
current in the wire be i, we have i = dQ/dt ; while Ohm's law,
modified by taking self-induction into account, gives the
equation
Eliminating i, we have
an equation which shows that when IFC < 4Z, the subsidence
of Q to zero is effected by oscillations of period
27T
(1- *
\LC 4Z
* Annales de Chiniie, xxxiv (1827), p. 5.
tProc. Am. Phil. Soc. ii (1842), p. 193.
J Phil. Mag. (4) v (1853), p. 400 ; Kelvin's Math, and Phys. Papers i,
p. 540.
254 The Mathematical Electricians of the
This simple result may be regarded as the beginning of the
theory of electric oscillations.
Thomson was at this time much engaged in the problems
of submarine telegraphy; and thus he was led to examine
the vexed question of the " velocity of electricity " over long
insulated wires and cables. Various workers had made
experiments on this subject at different times, but with
hopelessly discordant results. Their attempts had generally
taken the form of measuring the interval of time between the
appearance of sparks at two spark-gaps in the same circuit,
between which a great length of wire intervened, but which
were brought near each other in order that the discharges
might be seen together. In one series of experiments,
performed by Watson at Shooter's Hill in 1747-8,* the circuit
was four miles in length, two miles through wire and two
miles through the ground ; but the discharges appeared to be
perfectly simultaneous; whence Watson concluded that the
velocity of propagation of electric effects is too great to be
measurable.
In 1834 Charles Wheatstone,f Professor of Experimental
Philosophy in King's College, London, by examining in a
revolving mirror sparks formed a,t the extremities of a circuit,
found the velocity of electricity in a copper wire to be about
one and a half times the velocity of light. In 1850 H. Fizeau
and E. GounelleJ experimenting with the telegraph lines from
Paris to Eouen and to Amiens, obtained a velocity about one-
third that of light for the propagation of electricity in an iron
wire, and nearly two- thirds that of light for the propagation
in a copper wire.
The first step towards explaining these discrepancies was
made by Faraday, who§ early in 1854 showed experimentally
that a submarine cable, formed of copper wire covered with
* Phil. Trans, xlv (1748), pp. 49, 491.
t Phil. Trans., 1834, p. 583.
; Comptes Rendus, xxx (1850), p. 437.
§ Proc. Roy. Inst., Jan. 20, 1854: Phil. Mag-., June, 1854: Exp. Res. iii,
pp. 508, 521.
Middle of the Nineteenth Century. 255
gutta-percha, " may be assimilated exactly to an immense
Leyden battery ; the glass of the jars represents the gutta-
percha ; the internal coating is the surface of the copper wire,"
while the outer cgating corresponds to the sea-water. It
follows that in all calculations relating to the propagation of
electric disturbances along submarine cables, the electrostatic
capacity of the cable must be taken into account.
The theory of signalling by cable originated in a corre-
spondence between Stokes and Thomson in 1854. In the case
of long submarine lines, the speed of signalling is so much
limited by the electrostatic factor that electro-magnetic induc-
tion has no sensible effect ; and it was accordingly neglected in
the investigation. In view of other applications of the analysis;
however, we shall suppose that the cable has a self-induction L
per unit length, and that E denotes the ohmic resistance, and
C the capacity per unit length, Fthe electric potential at a
distance x from one terminal, and i the current at this place.
Ohm's law, as modified for inductance, is expressed by the
equation
9^ Tdi _>.
- -^- = L — + Ri ;
dx dt
moreover, since the rate of accumulation of charge in unit
length at # is - di/dx, and since this increases the potential
at the rate - (l/C^difix, we have
-
'dt dx
Eliminating i between these two equations, we have
1 82F
which is known as the equation of telegraphy*
Thomson, in one of his letterst to Stokes in 1854,
obtained this equation in the form which applies to Atlantic
.cables, i.e., with the term in L neglected. In this form it is
* "We have neglected leakage, which is beside our present purpose.
t Proc. Roy. Soc., May, 1855 : Kelvin's Math, and Phys. Papers, ii, p. 61.
256 The Mathematical Electricians of the
the same as Fourier's equation for the linear propagation of
heat : so that the known solutions of Fourier's theory may he
used in a new interpretation. If we substitute
v - /,2»<V - i -j- \x
y — t/ >
we obtain
A, = ± (1 + -v/^l) (nCR)l J
and therefore a typical elementary solution of the equation is
V = e-(nCR^x sin \2nt - (nCR)^x}.
The form of this solution shows that if a regular harmonic
variation of potential is applied at one end of a cable, the phase
is propagated with a velocity which is proportional to the
square root of the frequency of the oscillations : since therefore
the different harmonics are propagated with different velocities,
it is evident that no definite " velocity of transmission " is to be
expected for ordinary signals. If a potential is suddenly applied
at one end of the cable, a certain time elapses before the current
at the other end attains a definite percentage of its maximum
value ; but it may easily be shown* that this retardation is
proportional to the square of the length of the cable, so that
the apparent velocity of propagation would be less, the greater
the length of cable used.
The case of a telegraph line insulated in the air on poles is
different from that of a cable ; for here the capacity is small,
and it is necessary to take into account the inductance. If in
the general equation of telegraphy we write
V = enx^~l + Pl,
we obtain the equation
R (R* n* \i
2l f (±L* ~ CL) ;
as the capacity is small, we may replace the quantity under the
radical by its second term : and thus we see that a typical
elementary solution of the equation is
F= e i siu n{x - (CL)-1* t};
* This result, indeed, follows at once from the theory of dimensions.
Middle of the Nineteenth Century. 257
this shows that any harmonic disturbance, and therefore any
disturbance whatever, is propagated along the wire with
velocity (CL}~\ The difference between propagation in an
aerial wire and propagation in an oceanic cable is, as Thomson
remarked, similar to the difference between the propagation
of an impulsive pressure through a long column of fluid in a
tube when the tube is rigid (case of the aerial wire) and when
it is elastic, so as to be capable of local distension (case of the
cable, the distension corresponding to the effect of capacity) :
in the former case, as is well known, the impulse is propagated
with a definite velocity, namely, the velocity of sound in the
fluid.
The work of Thomson on signalling along cables was followed
in 1857 by a celebrated investigation* of Kirchhoff's, on the
propagation of electric disturbance along an aerial wire of
circular cross- section.
Kirchhoff assumed that the electric charge is practically all
resident on the surface of the wire, and that the current is
uniformly distributed over its cross-section; his idea of the
current was the same as that of Fechner and Weber, namely,
that it consists of equal streams of vitreous and resinous elec-
tricity flowing in opposite directions. Denoting the electric
potential by V, the charge per unit length of wire by e, the
length of the wire by I, and the radius of its cross-section by a,
he showed that Fis determined approximately by the equationf
V = 2e log (I/a).
* Ann. d. Phys. c (1857), pp. 193, 251 : Kirchhoff's Ges. Abhandl., p. iai ;
Phil. Mag. xiii (1857), p. 393.
t His method of obtaining this equation was to calculate separately the effects of
(1) the portion of the wire within a distance e on either side of the point con-
sidered, where e denotes a length small compared with J, but large compared with o,
and (2) the rest of the wire. He thus obtained the equation
where the integration is to be taken over all the length of the wire except the
portion 2e : the equation given in the text was then derived by an, approximation ,.
which, however, is open to some objection.
S
258 The Mathematical Electricians of the
The next factor to be considered is the mutual induction of
the current-elements in different parts of the wire. Assuming
with Weber that the electromotive force induced in an element
ds due to another element ds' carrying a current i' is derivable
from a vector-potential
,.3 •
Kirchhoff found for the vector-potential due to the entire wire
the approximate value
w = 2i log (//a),
where i denotes the strength of the current ;* the vector-
potential being directed parallel to the wire. Ohm's law then
gives the equation
ldw
where k denotes the specific conductivity of the material of
which the wire is composed; and finally the principle of
conservation of electricity gives the equation
di _ _de .
dx~ ~di'
Denoting log (I/a) by y, and eliminating e, i, w from these four
equations, we have
82F 1 d*V 1 8F
which is, as might have been expected, the equation of telegraphy.
When the term in 3 V/dt is ignored, as we have seen is in certain
cases permissible, the equation becomes
82F lF
* This expression was derived in a similar way to that for F, by an intermediate
formula
2 c ci'ds'
w = 2i log -- h — cos 6 cos Q ,
& a J r
where 6 and Q' denote respectively the angles made with r by ds and ds'.
Middle of the Nineteenth Century. 259
which shows that the electric disturbance is propagated along the
wire with the velocity c* KirchhofF s procedure has, in fact,
involved the calculation of the capacity and self-induction of
the wire, and is thus able to supply the definite values of the
quantities which were left undetermined in the general equation
of telegraphy.
The velocity c, whose importance was thus demonstrated, has
already been noticed in connexion with Weber's law of force ;
it is a factor of proportionality, which must be introduced when
electrodynamic phenomena are described in terms of units which
have been defined electrostatically ,f or conversely when units
which have been defined electrodynamicallyj are used in the
description of electrostatic phenomena. That the factor which
is introduced on such occasions must be of the dimensions
(length/time), may be easily seen : for the electrostatic re-
pulsion between electric charges is a quantity of the same kind
as the electrodynamic repulsion between two definite lengths of
wire, carrying currents which may be specified by the amount
of charge which travels past any point in unit time.
Shortly before the publication of Kirchhoff s memoir, the
value of c had been determined by Weber and Kohlrausch§ ;
their determination rested on a comparison of the measures of the
charge of a Leyden jar, as obtained by a method depending
on electrostatic attraction, and by a method depending on the
* In referring to the original memoirs of Weber and Kirchhoff, it must he
remembered that the quantity which in the present work is denoted by e, and
which represents the velocity of light in free aether, was by these writers denoted
by c/V'2. Weber, in fact, denoted by c the relative velocity with which two charges
must approach each other in order that the force between them, as calculated by
his formula, should vanish.
It must also be remembered that those writers who accepted the hypothesis
that currents consist of equal and opposite streams of vitreous and resinous
electricity, were accustomed to write 2t to denote the current-strength.
f i.e., defining unit electric charge as that which exerts unit ponderomotive
force on a conductor at unit distance which carries an equal charge ; and then
defining unit current as that which conveys unit charge in unit time.
% i.e., defining unit current by means of the ponderomotive force which it
exerts on an equal current, when the two currents flow in circuits of specified
form at a specified distance apart.
§ Ann. d. Phys. xcix (1856), p. 10.
S 2
260 The Mathematical Electricians of the
magnetic effects of the current produced by discharging the jar.
The resulting value was nearly
c = 3*1 x 1010 cm./sec.;
which was the same, within the limits of the errors of measure-
ment, as the speed with which light travels in interplanetary
space. The coincidence was noticed by Kirchhoff, who was thus
the first to discover the important fact that the velocity with
which an electric disturbance is propagated along a perfectly-
conducting aerial wire is equal to the velocity of light.
In a second memoir published in the same year, Kirchhoff*
extended the equations of propagation of electric disturbance
to the case of three-dimensional conductors.
As in his earlier investigation, he divided the electromotive
force at any point into two parts, of which one is the gradient
of the electrostatic potential </>, and the other is the derivate
with respect to the time (with sign reversed) of a vector-
potential a ; so that if i denote the current and k the specific
conductivity, Ohm's law is expressed by the equation
i = k (c2 grad <£ - a).
Kirchhoff calculated the value of a by aid of Weber's formula
for the inductive action of one current element on another;
the result is
where r denotes the vector from the point (x, y, z), at which a is
measured, to any other point (x, y, z") of the conductor, at which
the current is i' ; and the integration is extended over the whole
volume of the conductor. The remaining general equations are
the ordinary equation of the electrostatic potential
V2<£ + 4irp = 0
(where p denotes the density of electric charge), and the equation
of conservation of electricity
| + div i = 0.
ot
* Ann. d. Phys. cii (1857), p. 529 : Ges. AbhandL, p. 154.
Middle of the Nineteenth Century. 261
It will be seen that Kirchhoff's electrical researches were
greatly influenced by those of Weber. The latter investiga-
tions, however, did not enjoy unquestioned authority ; for there
was still a question as to whether the expressions given by
Weber for the mutual energy of two current elements, and for
the mutual energy of two electrons, were to be preferred to the
rival formulae of Neumann and Eiemann. The matter was
examined in 1870 by Helmholtz, in a series of memoirs* to
which reference has already been made.f Helmholtz remarked
that, for two elements ds, ds', carrying currents i, i', the electro-
dynamic energy is
n'(ds.ds')
r '
according to Neumann, and
?V
5-(r.ds)(r.ds'),
according to Weber; and that these expressions differ from
each other only by the quantity
- cos (ds . ds') + cos (r . ds) cos (r . ds') ] ,
dzr
or ^^
dsds
since this vanishes when integrated round either circuit, the
two formulae give the same result when applied to entire
currents. A general formula including both that of Neumann
and that of Weber is evidently
n'(ds .ds') .., ffr
— + ki^ -j—-, dsds,
r ds ds
where k denotes an arbitrary constant.^
Helmholtz's result suggested to Clausius§ a new form for
the law of force between electrons ; namely, that which is
* Journal fur Math., Ixxii (1870), p. 57 : Ixxv (1873), p. 35: Ixxviii (1874),
p. 273. t Cf. p. 229.
+ Cf. H. Lamb, Proc. Lond. Math. Soc., xiv (1883), p. 301.
§ Journal fiir Math. Ixxxii (1877), p. 85 : Phil. Mag., x (1880), p. 255.
262 The Mathematical Electricians of the
obtained by supposing that two electrons of charges e, e', and
velocities v, v', possess electrokinetic energy of amount
eef (v .v') 7 , d~r ,
— - - + kee -r— =-> w .
r dsds
Subtracting from this the mutual electrostatic potential energy,
which is ee'c'/r, we may write the mutual kinetic potential of
the two electrons in the form
(xx + ijy + zzf - c2) + kee' > vv',
where (x, y, z) denote the coordinates of e, and (X, y', z)
those of ef.
The unknown constant k has clearly no influence so long as
closed circuits only are considered: if k be replaced by zero,
the expression for the kinetic potential becomes
ee'
— (xx + yy + zz - c2),
which, as will appear later, closely resembles the corresponding
expression in the modern theory of electrons.
Clausius' formula has the great advantage over Weber's, that
it does not compel us to assume equal and opposite velocities
for the vitreous and resinous charges in an electric current;
on the other hand, Clausius' expression involves the absolute
velocities of the electrons, while Weber's depends only on their
relative motion; and therefore Clausius' theory requires the
assumption of a fixed aether in space, to which the velocities
v and V may be referred.
When the behaviour of finite electrical systems is predicted
from the formulae of Weber, Eiemann, and Clausius, the three
laws do not always lead to concordant results. For instance, if
a circular current be rotated with constant angular velocity
round its axis, according to Weber's law there would be a
development of free electricity on a stationary conductor in the
neighbourhood ; whereas, according to Clausius' formula there
would be no induction on a stationary body, but electrification
Middle of the Nineteenth Century. 263
would appear on a body turning with the circuit as if
rigidly connected with it. Again,* let a magnet be suspended
within a hollow metallic body, and let the hollow body be
suddenly charged or discharged; then, according to Clausius'
theory, the magnet is unaffected; but according to Weber's
and Kiemann's theories it experiences an impulsive couple.
And again, if an electrified disk be rotated in its own plane,
under certain circumstances a steady current will be induced in
a neighbouring circuit according to Weber's law, but not
according to the other formulae.
An interesting objection to Clausius' theory was brought
forward in 1879 by Frohlichf — namely, that when a charge of
free electricity and a constant electric current are at rest
relatively to each other, but partake together of the translatory
motion of the earth in space, a force should act between them if
Clausius' law were true. It was, however, shown by BuddeJ
that the circuit itself acquires an electrostatic charge, partly
as a result of the same action which causes the force on the
external conductor, and partly as a result of electrostatic
induction by the charge on the external conductor ; and that the
total force between the circuit and external conductor is thus
reduced to zero.§
We have seen that the discrimination between the different
laws of electrodynamic force is closely connected with the
question whether in an electric current there are two kinds of
electricity moving in opposite directions, or only one kind
moving in one direction. On the unitary hypothesis, that the
* The two following crudal experiments, with others, were suggested by
E. Budde, Ann. d. Phys. xxx (1887), p. 100.
t Ann. d. Phys. ix (1880), p. 261.
+ Ann. d. Phys. x (1880), p. 553.
§ This case of a charge and current moving side hy side was afterwards
examined by Fitz Gerald (Trans. Boy. Dub. Soc. i, 1882 ; Scient. Writings of
G. F. Fitz Gerald, p. Ill) without reference to Clausius' formula, from the
standpoint of Maxwell's theory. The result obtained was the same — namely,
that the electricity induced on the conductor carrying the current neutralizes the
ponderomotive force between the current and the external charge.
264 The Mathematical Electricians of the
current consists in a transport of one kind of electricity with a
definite velocity relative to the wire, it might be expected that
a coil rotated rapidly about its own axis would generate a
magnetic field different from that produced by the same coil
at rest. Experiments to determine the matter were performed
by A. Foppl* and by E. L. Nichols and W. S. Franklin,f but
with negative results. The latter investigators found that the
velocity of electricity must be such that the quantity conveyed
past a specified point in a unit of time, when the direction of
the current was that in which the coil was travelling, did not
differ from that transferred when the current and coil were
moving in opposite directions by as much as one part in ten
million, even when the velocity of the wire was 9096 cm./sec.
They considered that they would have been able to detect
a change of deflexion due to the motion of the coil, even though
the velocity of the current had been considerably greater than
a thousand million metres per second.
During the decades in the middle of the century consider-
able progress was made in the science of thermo-electricity,
whose beginnings we have already described. J In Faraday's
laboratory note-book, under the date July 28th, 1836, we
read§ : — " Surely the converse of thermo-electricity ought to be
obtained experimentally. Pass current through a circuit of
antimony and bismuth."
Unknown to Faraday, the experiment here indicated had
already been made, although its author had arrived at it by a
different train of ideas. In 1834 Jean Charles Peltier|| (b. 1785,
d. 1845) attempted the task, which was afterwards performed
with success by Joule,1J of measuring the heat evolved by the
passage of an electric current through a conductor. He found
that a current produces in a homogeneous conductor an elevation
* Ann. d. Phys. xxvii (1886), p. 410.
t Amer. Jour. Sci., xxxvii (1889), p. 103.
J Cf. pp. 92, 93. § Bence Jones's Life of Faraday, ii, p. 76.
II Annales de Ciiimie, Ivi (1834), p. 371. If Cf. p. 239.
Middle of the Nineteenth Century. 265
of temperature, which is the same in all parts of the conductor
where the cross-section is the same ; but he did not succeed in
connecting the thermal phenomena quantitatively with the
strength of .the current — a failure which was due chiefly to the
circumstance that his attention was fixed on the rise of
temperature rather than on the amount of the heat evolved.
But incidentally the investigation led to an important discovery
— namely, that when a current was passed in succession through
two conductors made of dissimilar metals, there was an evolution
of heat at the junction ; and that this depended on the direction of
the current ; for if the junction was heated when the current
flowed in one sense, it was cooled when the current flowed in the
opposite sense. This Peltier effect, as it is called, is quite distinct
from the ordinary Joulian liberation of heat, in which the
amount of energy set free in the thermal form is unaffected by
a reversal of the current ; the Joulian effect is, in fact, propor-
tional to the square of the current-strength, while the Peltier
effect is proportional to the current-strength directly. The
Peltier heat which is absorbed from external sources when a
current i flows for unit time through a junction from one metal
B to another metal A may therefore be denoted by
where T denotes the absolute temperature of the junction. The
function n^ (T) is found to be expressible as the difference of
two parts, of which one depends on the metal A only, and the
other on the metal B only ; thus we can write
In 1851 a general theory of thermo-electric phenomena was
constructed on the foundation of Seebeck's* and Peltier's dis-
coveries by W. Thomson.f Consider a circuit formed of two
* Cf. pp. 92, 93.
t Proc. R.S. Edinb. iii (1851), p. 91 ; Phil. Mag. iii (1852), p. 529 : Kelvin's
Math, and Phys. Paper*, i, p. 316. Cf. also Trans. R. S. Edinb. xxi (1854),
p. 123, reprinted in Papers, i, p. 232 : and Phil. Trans., 1856, reprinted in Papers,
ii, p. 189.
266 The Mathematical Electricians of the
metals, A and B, and let one junction be maintained at a
slightly higher temperature (T + $T) than the temperature T
of the other junction. As Seebeck had shown, a thermo-electric
current will be set up in the circuit. Thomson saw that such
a system might be regarded as a heat-engine, which absorbs a
certain quantity of heat at the hot junction, and converts part
of this into electrical energy, liberating the rest in the form of
heat at the cold junction. If the Joulian evolution of heat be
neglected, the process is reversible, and must obey the second
law of thermodynamics ; that is, the sum of the quantities of
heat absorbed, each divided by the absolute temperature at
which it is absorbed, must vanish. Thus we have
T+ST
so the Peltier effect H^(T) must be directly proportional to
the absolute temperature T. This result, however, as Thomson
well knew, was contradicted by the observations of Gumming,
who had shown that when the temperature of the hot junction
is gradually increased, the electromotive force rises to a maximum
value and then decreases. The contradiction led Thomson to
predict the existence of a hitherto unrecognized thermo-electric
phenomenon — namely, a reversible absorption of heat at places
in the circuit other than the junctions. Suppose that a current
flows along a wire which is of the same metal throughout, but
varies in temperature from point to point. Thomson showed
that heat must be liberated at some points and absorbed at
others, so as either to accentuate or to diminish the differences
of temperature at the different points of the wire. Suppose
that the heat absorbed from external sources when unit
electric charge passes from the absolute temperature T to the
temperature (T + $T) in a metal A is denoted by SA(T).ST.
The thermodynamical equation now takes the corrected form
~ SA(T)}
Middle of the Nineteenth Century. 267
Since the metals A and B are quite independent, this gives
This equation connects Thomson's " specific heat of electricity"
SA(T) with the Peltier effect.
In 1870 P. G. Tait* found experimentally that the specific
heat of electricity in pure metals is proportional to the absolute
temperature. We may therefore write SA(T) = aAT, where
a A denotes a constant characteristic of the metal A. The
thermodynamical equation then becomes
_d \UA(T))
dT ( T ~
or
where TTA denotes another constant characteristic of the metal.
The chief part of the Peltier effect arises from the term irAT.
By the investigations which have been described in the
present chapter, the theory of electric currents was considerably
advanced in several directions. In all these researches, how-
ever, attention was fixed on the conductor carrying the current
as the seat of the phenomenon. In the following period, interest
was centred not so much on the conductors which carry charges
and currents, as on the processes which take place in the
dielectric media .around them.
* Proc. R. S. Edinb. vii (1870), p. 308. Cf. also Batelli, Atti delia R. Ace. di
Torino, xxii (1886), p. 48, translated Phil. Mug. xxiv (1887), p. 295.
( 268 )
CHAPTEE VIII.
MAXWELL.
SINCE the time of Descartes, natural philosophers have never
ceased to speculate on the manner in which electric and
magnetic influences are transmitted through space. About
the middle of the nineteenth century, speculation assumed a
definite form, and issued in a rational theory.
Among those who thought much on the matter was Karl
Friedrich Gauss (b. 1777, d. 1855). In a letter* to Weber of
date March 19, 1845, Gauss remarked that he had long ago
proposed to himself to supplement the known forces which act
between electric charges by other forces, such as would cause
electric actions to be propagated between the charges with a
finite velocity. But he expressed himself as determined not
to publish his researches until he should have devised a
mechanism by which the transmission could be conceived to
be effected ; and this he had not succeeded in doing.
More than one attempt to realize Gauss's aspiration was
made by his pupil Eiemann. In a fragmentary note,t which
appears to have been written in 1853, but which was not
published until after his death, Biemann proposed an aether
whose elements should be endowed with the power of resisting
compression, and also (like the elements of MacCullagh's
aether) of resisting changes of orientation. The former pro-
perty he conceived to be the cause of gravitational and
electrostatic effects, and the latter to be the cause of optical
and magnetic phenomena. The theory thus outlined was
apparently not developed further by its author ; but in a short
investigation^ which was published posthumously in 1867,§ he
* Gauss' Werke, v, p. 629. t Riemann's Werke, 2e Aufl., p. 526.
J Ann. d. Phys. cxxxi (1867), p. 237 ; Riemann's Werke, 2e Aufl., p. 288 ;
Phil. Mag. xxxiv (1867), p. 368.
§ It had been presented to the Gottingen Academy in 1858, but afterwards
withdrawn.
.
Maxwell. 269
returned to the question of the process by which electric action
is propagated through space. In this memoir he proposed to
replace Poisson's equation for the electrostatic potential,
namely,
by the equation
according to which the changes of potential due to changing
electrification would be propagated outwards from the charges
with a velocity c. This, so far as it goes, is in agreement with
the view which is now accepted as correct ; but Kiemann's
hypothesis was too slight to serve as the basis of a complete
theory. Success came only when the properties of the inter-
vening medium were taken into account.
In that power to which Gauss attached so much importance,
of devising dynamical models and analogies for obscure physical
phenomena, perhaps no one has ever excelled W. Thomson*;
and to him, jointly with Faraday, is due the credit of having
initiated the theory of the electric medium. In one of his
earliest papers, written at the age of seventeen,! Thomson
compared the distribution of electrostatic force, in a region
containing electrified conductors, with the distribution of the
flow of heat in an infinite solid : the equipotential surfaces in
the one case correspond to the isothermal surfaces in the other,
and an electric charge corresponds to a source of heat.J
* As will appear from the present chapter, Maxwell had the same power in a
very marked degree. It has always been cultivated hy the " Cambridge school "
of natural philosophers.
t Camb. Math. Journal, iii (Feb. 1842), p. 71 ; reprinted in Thomson's Papers
<JH Electrostatics and Magnetism, p. 1. Also Camb. and Dub. Math. Journal,
Nov., 1845 ; reprinted in Papers, p. 15.
\ As regards this comparison, Thomson had been anticipated by Chasles,
Journal de 1'Ec. Polyt. xv (1837), p. 266, who had shown that attraction accord-
ing to Newton's law gives rise to the same fields as the steady conduction of heat,
both depending on Laplace's equation v' V =• 0.
It will be remembered that Ohm had used an analogy between thermal conduction
and galvanic phenomena.
270 Maxwell.
It may, perhaps, seem as if the value of such an analogy
as this consisted merely in the prospect which it offered of
comparing, and thereby extending, the mathematical theories
of heat and electricity. But to the physicist its chief interest
lay rather in the idea that formulae which relate to the electric
field, and which had heen deduced from laws of action at a
distance, were shown to be identical with formulae relating to
the theory of heat, which had been deduced from hypotheses
of action between contiguous particles.
In 1846 — the year after he had taken his degree as second
wrangler at Cambridge — Thomson investigated* the analogies
of electric phenomena with those of elasticity. For this purpose
he examined the equations of equilibrium of an incompressible
elastic solid which is in a state of strain ; and showed that
the distribution of the vector which represents the elastic
displacement might be assimilated to the distribution of the
electric force in an electrostatic system. This, however, as he
went on to show, is not the only analogy which may be
perceived with the equations of elasticity ; for the elastic
displacement may equally well be identified with a vector a,
defined in terms of the magnetic induction B by the relation
curl a = B.
The vector a is equivalent to the vector-potential which
had been used in the memoirs of Neumann, Weber, and
Kirchhoff, on the induction of currents ; but Thomson arrived
at it independently by a different process, and without being at
the time aware of the identification.
The results of Thomson's memoir seemed to suggest a
picture of the propagation of electric or magnetic force : might
it not take place in somewhat the same way as changes in the
elastic displacement are transmitted through an elastic solid ?
These suggestions were not at the time pursued further
by their author; but they helped to inspire another young
* Camb. and Dub. Math. Journ. ii (1847), p. 61 : Thomson's Math, and Phys.
Papers, i, p. 76.
Maxwell. 271
Cambridge man to take up the matter a few years later.
James Clerk Maxwell, by whom the problem was eventually
solved, was born in 1831, the son of a landed proprietor in
Dumfriesshire. He was educated at Edinburgh, and at Trinity
College, Cambridge, of which society he became in 1855 a
Fellow; and not long after his election to Fellowship, he
communicated to the Cambridge Philosophical Society the first
of his endeavours* to form a mechanical conception of the
electro-magnetic field.
Maxwell had been reading Faraday's Experimental He-
searches', and, gifted as he was with a physical imagination
akin to Faraday's, he had been profoundly impressed by the
theory of lines of force. At the same time, he was a trained
mathematician ; and the distinguishing feature of almost all
his researches was the union of the imaginative and the
analytical faculties to produce results partaking of both
natures. This first memoir may be regarded as an attempt to
connect the ideas of Faraday with the mathematical analogies
which had been devised by Thomson.
Maxwell considered first the illustration of Faraday's lines
of force which is afforded by the lines of flow of a liquid. The
lines of force represent the direction of a vector; and the
magnitude of this vector is everywhere inversely proportional
to the cross-section of a narrow tube formed by such lines.
This relation between magnitude and direction is possessed by
any circuital vector ; and in particular by the vector which
represents the velocity at any point in a fluid, if the fluid be
incompressible. It is therefore possible to represent the
magnetic induction B, which is the vector represented by
Faraday's lines of magnetic force, as the velocity of an incom-
pressible fluid. Such an analogy had been indicated some
years previously by Faraday himself,f who had suggested that
along the lines of magnetic force there may be a " dynamic
condition," analogous to that of the electric current, and
* Trans. Camb. Phil. Soc. x, p. 27; Maxwell's Scientific Papers, i, p. 155.
t Exp. Res., § 3269 (1852).
272 Maxwell.
that, in fact, " the physical lines of magnetic force are
currents."
The comparison with the lines of flow of a liquid is
applicable to electric as well as to magnetic lines of force. In
this case the vector which corresponds to the velocity of the
fluid is, in free aether, the electric force E. But when different
dielectrics are present in the field, the electric force is not a
circuital vector, and, therefore cannot be represented by lines
of force ; in fact, the equation
div E = 0
is now replaced by the equation
div(eE) = 0,
where g denotes the specific inductive capacity or dielectric
constant at the place (x, y} z\ It is, however, evident from
this equation that the vector cE is circuital ; this vector,
which will be denoted by D, bears to E a relation similar to
that which the magnetic induction B bears to the magnetic
force H. It is the vector D which is represented by Faraday's
lines of electric force, and which in the hydrodynamical
analogy corresponds to the velocity of the incompressible fluid.
In comparing fluid motion with electric fields it is necessary
to introduce sources and sinks into the fluid to correspond to
the electric charges ; for D is not circuital at places where there,
is free charge. The magnetic analogy is therefore somewhat
the simpler.
In the latter half of his memoir Maxwell discussed how
Faraday's "electrotonic state" might be represented in mathe-
matical symbols. This problem he solved by borrowing from
Thomson's investigation of 1847 the vector a, which is defined
in terms of the magnetic induction by the equation
curl a = B ;
if, with Maxwell, we call a the electrotonic intensity, the.
equation is equivalent to the statement that " the entire
electrotonic intensity round the boundary of any surface
measures the number of lines of magnetic force which pass,
Maxwell. 273
through that surface." The electromotive force of induction at
the place (x, y, z) is - d&/dt : as Maxwell said, " the electromotive
force on any element of a conductor is measured by the
instantaneous rate of change of the electrotonic intensity on
that element." From this it is evident that a is no other than
the vector-potential which had been employed by Neumann,
Weber, and Kirchhoff, in the calculation of induced currents ;
and we may take* for the electrotonic intensity due to a
current ir flowing in a circuit s' the value which results from
Neumann's theory, namely,
., f *s'
= t'
} r
It may, however, be remarked that the equation
curl a = B,
taken alone, is insufficient to determine a uniquely ; for we can
choose a so as to satisfy this, and also to satisfy the equation
div a = ;//,
where i// denotes any arbitrary scalar. There are, therefore, an
infinite number of possible functions a. With the particular
value of a which has been adopted, we have
3 ., f dx' 8 f dy' 8 ., f dz
div a = - i \ - + — ^' -2- + - i' \ —
te I' r fy )8, r dz J, r
.,
*« ¥
= 0;
so the vector-potential a which we have chosen is circuital.
In this memoir the physical importance of the operators
curl and div first became evidentf ; for, in addition to those
applications which have been mentioned, Maxwell showed that
* Cf . p. 224.
t These operators had, however, occurred frequently in the writings of Stokes
especially in his memoir of 1849 on the Dynamical Theory of Diffraction.
T
274 Maxwell.
he connexion between the strength i of a current and the
magnetic field H, to which it gives rise, may be represented by
the equation
4?ri = curl H ;
this equation is equivalent to the statement that " the entire
magnetic intensity round the boundary of any surface measures
the quantity of electric current which passes through that
surface."
In the same year (1856) in which Maxwell's investigation
was published, Thomson* put forward an alternative inter-
pretation of magnetism. He had now come to the conclusion,
from a study of the magnetic rotation of the plane of polariza-
tion of light, that magnetism possesses a rotatory character;
and suggested that the resultant angular momentum of the
thermal motions of a bodyf might be taken as the measure of
the magnetic moment. " The explanation," he wrote, " of all
phenomena of electromagnetic attraction or repulsion, or of
electromagnetic induction, is to be looked for simply in the
inertia or pressure of the matter of which the motions
constitute heat. Whether this matter is or is not electricity,
whether it is a continuous fluid interpermeating the spaces
between molecular nuclei, or is itself molecularly grouped : or
whether all matter is continuous, and molecular heterogeneous-
ness consists in finite vortical or other relative motions of
contiguous parts of a body: it is impossible to decide, and,
perhaps, in vain to speculate, in the present state of science."
The two interpretations of magnetism, in which the linear
and rotatory characters respectively are attributed to it, occur
frequently in the subsequent history of the subject. The
former was amplified in 1858, when Helmholtz published his
researches^ on vortex motion ; for Helmholtz showed that if a
*Proc. Roy. Soc. viii (1856), p. 150 ; xi (1861), p. 327, foot-note: Phil. Mag.
xiii (1857), p. 198; Baltimore Lectures, Appendix F.
t This was written shortly before the kinetic theory of gases was developed
by Clausius and Maxwell.
+ Journal fur Math. Iv (1858), p. 25; Helmholtz's Wiss. Abh. i, p. 101;
translated Phil. Mag. xxxiii (1867), p. 485.
Harwell. 275
magnetic field produced by electric currents is compared to the
flow of an incompressible fluid, so that the magnetic vector is
represented by the fluid velocity, then the electric currents
correspond to the vortex-filaments in the fluid. This analogy
correlates many theorems in hydrodynamics and electricity ;
for instance, the theorem that a re-entrant vortex-filament is
equivalent to a uniform distribution of doublets over any
surface bounded by it, corresponds to Ampere's theorem of the
equivalence of electric currents and magnetic shells.
In his memoir of 1855, Maxwell had not attempted to
construct a mechanical model of electrodynamic actions, but
had expressed his intention of doing so. " By a careful study,"
he wrote,* " of the laws of elastic solids, and of the motions of
viscous fluids, I hope to discover a method of forming a
mechanical conception of this electrotonic state adapted to
general reasoning " ; and in a foot-note he referred to the effort
which Thomson had already made in this direction. Six years
elapsed, however, before anything further on the subject was
published. In the meantime, Maxwell became Professor of
Natural Philosophy in King's College, London — a position in
which he had opportunities of personal contact with Faraday,
whom he had long reverenced. Faraday had now concluded
the Experimental Researches, and was living in retirement at
Hampton Court ; but his thoughts frequently recurred to the
great problem which he had brought so near to solution. It
appears from his note-book that in 1857f he was speculating
whether the velocity of propagation of magnetic action is of the
same order as that of light, and whether it is affected by the
susceptibility to induction of the bodies through which the
action is transmitted.
The answer to this question was furnished in 1861-2,
when Maxwell fulfilled his promise of devising a mechanical
conception of the electromagnetic field.*
* Maxwell's Scientific Papers, i, p. 188.
t Bence Jones's Life of Faraday ii, p. 379.
I Phil. Mag. xxi (1861), pp. 161, 281, 338; xxiii (1862), pp. 12, 85;
Maxwell's Scientific Papers, i, p. 451.
T 2
276 Maxwell.
In the interval since the publication of his previous memoir
Maxwell had become convinced by Thomson's arguments that
magnetism is in its nature rotatory. "The transference of
electrolytes in fixed directions by the electric current, and the
rotation of polarized light in fixed directions by magnetic force,
are," he wrote, "the facts the consideration of which has
induced me to regard magnetism as a phenomenon of rotation,
and electric currents as phenomena of translation." This con-
ception of magnetism he brought into connexion with Faraday's
idea, that tubes of force tend to contract longitudinally and to
expand laterally. Such a tendency may be attributed to
centrifugal force, if it be assumed that each tube of force
contains fluid which is in rotation about the axis of the tube.
Accordingly Maxwell supposed that, in any magnetic field, the
medium whose vibrations constitute light is in rotation about
the lines of magnetic force; each unit tube of force may for the
present be pictured as an isolated vortex.
The energy of the motion per unit volume is proportional
to /jH2, where /j. denotes the density of the medium, and H
denotes the linear velocity at the circumference of each vortex.
But, as we have seen,* Thomson had already shown that the
energy of any magnetic field, whether produced by magnets or
by electric currents, is
where the integration is taken over all space, and where it
denotes the magnetic permeability, and H the magnetic force.
It was therefore natural to identify the density of the medium
at any place with the magnetic permeability, and the circum-
ferential velocity of the vortices with the magnetic force.
But an objection to the proposed analogy now presents
itself. Since two neighbouring vortices rotate in the same
direction, the particles in the circumference of one vortex must
be moving in the opposite direction to the particles contiguous
* Cf. pp. 248, 250.
Maxwell. 277
to them in the circumference of the adjacent vortex ; and it
seems, therefore, as if the motion would be discontinuous.
Maxwell escaped from this difficulty by imitating a well-known
mechanical arrangement. When it is desired that two wheels
should revolve in the same sense, an " idle " wheel is inserted
between them so as to be in gear with both. The model of the
electromagnetic field to which Maxwell arrived by the intro-
duction of this device greatly resembles that proposed by
Bernoulli in 1736.* He supposed a layer of particles, acting as
idle wheels, to be interposed between each vortex and the next,
and to roll without sliding on the vortices ; so that each vortex
tends to make the neighbouring vortices revolve in the same
direction as itself. The particles were supposed to be not other-
wise constrained, so that the velocity of the centre of any
particle would be the mean of the circumferential velocities of
the vortices between which it is placed. This condition yields
(in suitable units) the analytical equation
47Ti = curl H,
where the vector i denotes the flux of the particles, so that its
^-component ix denotes the quantity of particles transferred
in unit time across unit area perpendicular to the ^-direction.
On comparing this equation with that which represents Oersted's
discovery, it is seen that the flux i of the movable particles
interposed between neighbouring vortices is the analogue of
the electric current.
It will be noticed that in Maxwell's model the relation
between electric current and magnetic force is secured by a
connexion which is not of a dynamical, but of a purely kine-
matical character. The above equation simply expresses the
existence of certain non-holonomic constraints within the
system.
If from any cause the rotatory velocity of some of the
cellular vortices is altered, the disturbance will be propagated
from that part of the model to all other parts, by the mutual
* Cf. p. 100.
278 Maxwell.
action of the particles and vortices. This action is determined,
as Maxwell showed, hy the relation
fj$L = - curl E
which connects E, the force exerted on a unit quantity of
particles at any place in consequence of the tangential action
of the vortices, with H, the rate of change of velocity of the
neighbouring vortices. It will be observed that this equation
is not kinematical but dynamical. On comparing it with the
electromagnetic equations
curl a = /*H,
Induced electromotive force = - a,
it is seen that E must be interpreted electromagnetically as the
induced electromotive force. Thus the motion of the particles
constitutes an electric current, the tangential force with which
they are pressed by the matter of the vortex-cells constitutes
electromotive force, and the pressure of the particles on each
other may be taken to correspond to the tension or potential of
the electricity.
The mechanism must next be extended so as to take account
of the phenomena of electrostatics. For this purpose Maxwell
assumed that the particles, when they are displaced from their
equilibrium position in any direction, exert a tangential action
on the elastic substance of the cells ; and that this gives rise
to a distortion of the cells, which in turn calls into play a
force arising from their elasticity, equal and opposite to the
force which urges the particles away from the equilibrium
position. When the exciting force is removed, the cells recover
their form, and the electricity returns to its former position.
The state of the medium, in which the electric particles are
displaced in a definite direction, is assumed to represent an
electrostatic field. Such a displacement does not itself con-
stitute a current, because when it has attained a certain value
it remains constant ; but the variations of displacement are to
be regarded as currents, in the positive or negative direction
according as the displacement is increasing or diminishing.
Maxwell. 279
The conception of the electrostatic state as a displacement
of something from its equilibrium position was not altogether
new, although it had not been previously presented in this
form. Thomson, as we have seen, had compared electric force
to the displacement in an elastic solid ; and Faraday, who had
likened the particles of a ponderable dielectric to small con-
ductors embedded in an insulating medium,* had supposed that
when the dielectric is subjected to an electrostatic field, there
is a displacement of electric charge on each of the small
conductors. The motion of these charges, when the field is
varied, is equivalent to an electric current ; and it was from
this precedent that Maxwell derived the principle, which became
of cardinal importance in his theory, that variations of displace-
ment are to be counted as currents. But in adopting the
idea, he altogether transformed it ; for Faraday's conception of
displacement was applicable only to ponderable dielectrics, and
was in fact introduced solely in order to explain why the
specific inductive capacity of such dielectrics is different from
that of free aether; whereas according to Maxwell there is
displacement wherever there is electric force, whether material
bodies are present or not.
The difference between the conceptions of Faraday and
Maxwell in this respect may be illustrated by an analogy
drawn from the theory of magnetism. When a piece of iron
is placed in a magnetic field, there is induced in it a magnetic
distribution, say of intensity I ; this induced magnetization
exists only within the iron, being zero in the free aether
outside. The vector I may be compared to the polarization
or displacement, which according to Faraday is induced in
dielectrics by an electric field; and the electric current con-
stituted by the variation of this polarization is then analogous
to dl/dt. But the entity which was called by Maxwell the
electric displacement in the dielectric is analogous not to I,
but to the magnetic induction B : the Maxwellian displace-
* Cf. p. 210.
280 Maxwell.
merit-current corresponds to d'B/dt, and may therefore have a
value different from zero even in free aether.
It may be remarked in passing that the term displacement,
which was thus introduced, and which has been retained in
the later development of the theory, is perhaps not well chosen ;
what in the early models of the aether was represented as an
actual displacement, has in later investigations been conceived
of as a change of structure rather than of position in the
elements of the aether.
Maxwell supposed the electromotive' force acting on the
electric particles to be connected with the displacement D
which accompanies it, by an equation of the form
where c, denotes a constant which depends on the elastic
properties of the cells. The displacement-current D must now
be inserted in the relation which connects the current with
the magnetic force ; and thus we obtain the equation
curl H = 47rS,
where the vector S, which is called the total current, is the
sum of the convection-current i and the displacement-current
D. By performing the operation div on both sides of this
equation, it is seen that the total current is a circuital vector.
In the model, the total current is represented by the total
motion of the rolling particles ; and this is conditioned by the
rotations of the vortices in such a way as to impose the
kinematic relation
div S = 0.
Having obtained the equations of motion of his system
of vortices and particles, Maxwell proceeded to determine the
rate of propagation of disturbances through it. He considered
in particular the case in which the substance represented is a
dielectric, so that the conduction-current is zero. If, moreover,
Maxwell. 281
the constant fi be supposed to have the value unity, the
equations may be written
div H = 0,
c,2 curl H = E,
- curl E = H.
Eliminating E, we see* that H satisfies the equations
jdivH = 0,
•«•
But these are precisely the equations which the light- vector
satisfies in a medium in which the velocity of propagation is c^ :
it follows that disturbances are propagated through the model
by waves which are similar to waves of light, the magnetic
(and similarly the electric) vector being in the wave-front.
For a plane-polarized wave propagated parallel to the axis of z,
the equations reduce to
2y = x 2*^y y
"Cl dz '"' dt' Cl ~dz '' dt' dz dt' dz
whence we have
= Ex - c\Sx = E
these equations show that the electric and magnetic vectors are
at right angles to each other.
The question now arises as to the magnitude of the constant
Cj.f This may be determined by comparing different expressions
for the energy of an electrostatic field. The work done by an
electromotive force E in producing a displacement D is
fD
E . dD or JED
o
per unit volume, since E is proportional to D. But if it be
assumed that the energy of an electrostatic field is resident in
the dielectric, the amount of energy per unit volume may be
* For if a denote any vector, we have identically
V-a -f grad div a + curl curl a = 0.
t For criticisms on the procedure by which Maxwell determined the velocity of
propagation of disturbance, cf. P. Duhem, Les Theories Electriqv.es de J. Clerk
Maxwell, Paris, 1902.
282 Maxwell.
calculated by considering the mechanical force required in
order to increase the distance between the plates of a condenser,
so as to enlarge the field comprised between them. The result
is that the energy per unit volume of the dielectric is fE/2/87r,
where c denotes the specific inductive capacity of the dielectric
and E' denotes the electric force, measured in terms of the
electrostatic unit : if E denotes the electric force expressed in
terms of the electrodynamic units used in the present investi-
gation, we have E = cE', where c denotes the constant which*
occurs in transformations of this kind. The energy is therefore
fcE2/87TC2 per unit volume. Comparing this with the expression
for the energy in terms of E and D, we have
D
and therefore the constant Ci has the value ct*. Thus the
result is obtained that the velocity of propagation of dis-
turbances in Maxwell's medium is ce~£, where £ denotes the
specific inductive capacity and c denotes the velocity for which
Kohlrausch and Weber had foundf the value 3*1 x 1010 cm./sec.
Now by this time the velocity of light was known, not only
from the astronomical observations of aberration and of Jupiter's
satellites, but also by direct terrestrial experiments. In 1849
Hippolyte Louis FizeauJ had ' determined it by rotating a
toothed wheel so rapidly that a beam of light transmitted
through the gap between two teeth and reflected back from a
mirror was eclipsed by one of the teeth on its return journey.
The velocity of light was calculated from the dimensions and
angular velocity of the wheel and the distance of the mirror ;
the result being 3*15 x 1010 cm. /sec. §
* Cf. pp. 227, 259. | Cf. p. 260.
| Comptes Rendus, xxix (1849), p. 90. A determination made by Cornu in
1874 was on this principle.
§ A different experimental method was employed in 1862 hy Leon Foucault
(Comptes Rendus, Iv, pp. 501, 792) ; in this a ray from an origin 0 was reflected
by a revolving mirror M to a fixed mirror, and so reflected back to J/, and again
to O. It is evident that the returning ray ?dO must be deviated by twice the
angle through which M turns while the light passes from M to the fixed mirror
and back. The value thus obtained by Foucault for the velocity of light was
Maxwell. 283
Maxwell was impressed, as Kirchhoff had been before him,
by the close agreement between the electric ratio c and the
velocity of light* ; and having demonstrated that the propaga-
tion of electric disturbance resembles that of light, he did not
hesitate to assert the identity of the two phenomena. "We
can scarcely avoid the inference," he said, " that light consists
in the transverse undulations of the same medium which is the
cause of electric and magnetic phenomena." Thus was answered
the question which Priestley had asked almost exactly a hundred
years before :f "Is there any electric fluid sui generis at all,
distinct from the aether ? "
The presence of the dielectric constant e in the expression
ct -i, which Maxwell had obtained for the velocity of propaga-
tion of electromagnetic disturbances, suggested a further test
of the identity of these disturbances with light: for the velocity
of light in a medium is known to be inversely proportional to
the refractive index of the medium, and therefore the refractive
index should be, according to the theory, proportional to the
square root of the specific inductive capacity. At the time,
however, Maxwell did not examine whether this relation
was confirmed by experiment.
In what has preceded, the magnetic permeability //, has been
supposed to have the value unity. If this is not the case, the
2-98 x 1010 cm./sec. Subsequent determinations by Michelson in 187'.) (Ast.
Papers of the Amer. Ephemeris, i), and by Newcomb in 1882 (ibid., ii) depended
on the same principle.
As was shown afterwards by Lord Rayleigh (Nature, xxiv, p. 382, xxv, p. 52)
and by Gibbs (Nature, xxxiii, p. 582), the value obtained for the velocity of light
by the methods of Fizeau and Foucault represents the group-velocity, not the wave-
velocity ; the eclipses of Jupiter's satellites also give the group-velocity, while the
value deduced from the coefficient of aberration is the wave- velocity. In a non-
dispersive medium, the group- velocity coincides with the wave- velocity ; and the
agreement of the values of the velocity of light obtained by the two astronomical
methods seems to negative the possibility of any appreciable dispersion in free
aether.
The velocity of light in dispersive media was directly investigated by Michelson
in 1883-4, with results in accordance with theory.
* He had "worked out the formulae in the country, before seeing Weber's
result." Cf. Campbell and Garnett's Life of Maxwell, p. 244.
f Priestley's Eistory, p. 488.
284 Maxwell.
velocity of propagation of disturbance may be shown, by the
same analysis, to be ct~i^~i ; so that it is diminished when /u is
greater than unity, i.e., in paramagnetic bodies. This inference
had been anticipated by Faraday : " Nor is it likely," he wrote,*
" that the paramagnetic body oxygen can exist in the air and
not retard the transmission of the magnetism."
It was inevitable that a theory so novel and so capacious as
that of Maxwell should involve conceptions which his contempo-
raries understood with difficulty and accepted with reluctance.
Of these the most difficult and unacceptable was the principle
that the total current is always a circuital vector ; or, as it is
generally expressed, that " all currents are closed." According
to the older electricians, a current which is employed in charging
a condenser is not closed, but terminates at the coatings of the
condenser, where charges are accumulating. Maxwell, on the
other hand, taught that the dielectric between the coatings
is the seat of a process — the displacement-current — which is
proportional to the rate of increase of the electric force in the
dielectric ; and that this process produces the same magnetic
effects as a true current, and forms, so to speak, a continuation,
through the dielectric, of the charging current, so that the
latter may be regarded as flowing in a closed circuit.
Another characteristic feature of Maxwell's theory is the
conception — for which, as we have seen, he was largely indebted
to Faraday and Thomson — that magnetic energy is the kinetic
energy of a medium occupying the whole of space, and that
electric energy is the energy of strain of the same medium.
By this conception electromagnetic theory was brought into
such close parallelism with the elastic- solid theories of the
aether, that it was bound to issue in an electromagnetic theory
of light.
Maxwell's views were presented in a more developed form
in a memoir entitled "A Dynamical Theory of the Electro-
magnetic Field," which was read to the Koyal Society in 1864 ;f
* Faraday's laboratory note-book for 1857 : of. Bence Jones's Life of Faraday,
ii, p. 380.
t Phil. Trans, civ (1865), p. 459 : Maxwell's Scient. Papers, i, p. 526
Maxwell. 285
in this the architecture of his system was displayed, stripped of
the scaffolding by aid of which it had been first erected.
As the equations employed were for the most part the same
as had been set forth in the previous investigation, they need
only be briefly recapitulated. The magnetic induction juH, being
a circuital vector, may be expressed in terms of a vector-potential
A by the equation
luiK = curl A.
The electric displacement D is connected with the volume-
density p of free electric charge by the electrostatic equation
div D = p.
The principle of conservation of electricity yields the equation
div i = - dp/dt,
where i denotes the conduction-current.
The law of induction of currents — namely, that the total
electromotive force in any circuit is proportional to the rate of
decrease of the number of lines of magnetic induction which
pass through it — may be written
- curl E = /LtH ;
from which it follows that the electric force E must be expressible
in the form
E = - A + grad i//,
where ^ denotes some scalar function. The quantities A and ;//
which occur in this equation are not as yet completely deter-
minate ; for the equation by which A is defined in terms of the
magnetic induction specifies only the circuital part of A ; and as
the irrotational part of A is thus indeterminate, it is evident
that \p also must be indeterminate. Maxwell decided the matter
by assuming* A to be a circuital vector ; thus
divA = 0,
and therefore div E = -
* This is the effect of the introduction of (F1, G', H'} in § 98 of the memoir ;
cf. also Maxwell's Treitise on Electricity and Magnetism, § 616.
286 Maxwell.
from which equation it is evident that ^ represents the electro-
static potential.
The principle which is peculiar to Maxwell's theory must
now be introduced. Currents of conduction are not the only
kind of currents ; even in the older theory of Faraday, Thomson,
and Mossotti, it had been assumed that electric charges
are set in motion in the particles of a dielectric when the
dielectric is subjected to an electric field ; and the prede-
cessors of Maxwell would not have refused to admit that the
motion of these charges is in some sense a current. Suppose,
then, that S denotes the total current which is capable of
generating a magnetic field : since the integral of the magnetic
force round any curve is proportional to the electric current
which flows through the gap enclosed by the curve, we have in
suitable units
curl H = 4;rS.
In order to determine S, we may consider the case of a con-
denser whose coatings are supplied with electricity by a
conduction-current i per unit-area of coating. If ± o- denote
the surface-density of electric charge on the coatings, we have
i = d(r/dtt and o- = D,
where D denotes the magnitude of the electric displacement D
in the dielectric between the coatings ; so i = D. But since the
total current is to be circuital, its value in the dielectric must
be the same as the value i which it has in the rest of the
circuit ; that is, the current in the dielectric has the value D.
We shall assume that the current in dielectrics always has this
value, so that in the general equations the total current must
be understood to be i + D.
The above equations, together with those which express the
proportionality of E to D in insulators, and to i in conductors,
constituted Maxwell's system for a field formed by isotropic
bodies which are not in motion. When the magnetic field is
.due entirely to currents (including both conduction-currents
Maxwell. 287
and displacement-currents), so that there is no magnetization,
we have
V2A = - curl curl A = - curl H
= - 47TS,
so that the vector-potential is connected with the total current
by an equation of the same form as that which connects the
scalar potential with the density of electric charge. To these
potentials Maxwell inclined to attribute a physical significance ;
he supposed i// to be analogous to a pressure subsisting in the
mass of particles in his model, and A to be the measure of
the electrotonic state. The two functions are, however, of
merely analytical interest, and do not correspond to physical
entities. For let two oppositely-charged conductors, placed
close to each other, give rise to an electrostatic field throughout
all space. In such a field the vector-potential A is everywhere
zero, while the scalar potential $ has a definite value at every
point. Now let these conductors discharge each other ; the
electrostatic force at any point of space remains unchanged
until the point in question is reached by a wave of disturbance,
which is propagated outwards from the conductors with the
velocity of light, and which annihilates the field as it passes
over it. But this order of events is not reflected in the
behaviour of Maxwell's functions ;// and A ; for at the instant
of discharge, ^ is everywhere annihilated, and A suddenly
acquires a finite value throughout all space.
As the potentials do not possess any physical significance,
it is desirable to remove them from the equations. This was
afterwards done by Maxwell himself, who* in 1868- proposed
to base the electromagnetic theory of light solely on the
equations
curl H = 47rS,
- curl E = B,
together with the equations which define S in terms of E, and B
in terms of H.
* Phil. Trans, clviii (1868), p. 643 : Maxwell's Scient. Papers, ii, p. 125.
288 Maxwell.
The memoir of 1864 contained an extension of the equations
to the case of bodies in motion ; the consideration of which
naturally revives the question as to whether the aether is in
any degree carried along with a body which moves through it.
Maxwell did not formulate any express doctrine on this subject ;
but his custom was to treat matter as if it were merely a
modification of the aether, distinguished only by altered
values of such constants as the magnetic permeability and
the specific inductive capacity ; so that his theory may be
said to involve the assumption that matter and aether move
together. In deriving the equations which are applicable to
moving bodies, he made use of Faraday's principle that the
electromotive force induced in a body depends only on the
relative motion of the body and the lines of magnetic force,
whether one or the other is in motion absolutely. From this
principle it may be inferred that the equation which determines
the electric force* in terms of the potentials, in the case of a
body which is moving with velocity w, is
E = [w . /zH] - A + grad ^.
Maxwell thought that the scalar quantity -fy in this equation
represented the electrostatic potential; but the researches of
other investigators-)- have indicated that it represents the sum
of the electrostatic potential and the quantity (A . w).
The electromagnetic theory of light was moreover extended
in this memoir so as to account for the optical properties of
crystals. For this purpose Maxwell assumed that in crystals
the values of the coefficients of electric and magnetic induction
depend on direction, so that the equation
fjbK = curl A
is replaced by
= curl A ;
* It may be here remarked that later writers have distinguished between the
electric force in a moving body and the electric force in the aether through which
the body is moving, and that E in the present equation corresponds to the former
of these vectors.
t Helmholtz, Journ. fiir Math., Ixxviii (1874), p. 309; H. W. Watson, Phil.
Mag. (5), xxv (1888), p. 271.
Maxwell. 289
and similarly the equation
E = 47rcO>/6
is replaced by
E = 4;r (c?D.xt c?Dy, cjDz\
The other equations are the same as in isotropic media ; so that
the propagation of disturbance is readily seen to depend on the
equation
(/i J?» ft.ffy, HZHZ} = - curl [c,2 (curl 5),, tf(cuilH}y, Ca2 (curl -#)*)•
Now, if jui, ju2, A*3 are supposed equal to each other, this
equation is the same as the equation of motion of MacCullagh's
aether in crystalline media,* the magnetic force H corresponding
to MacCullagh's elastic displacement ; and we may therefore
immediately infer that Maxwell's electromagnetic equations
yield a satisfactory theory of the propagation of light in
crystals, provided it is assumed that the magnetic permeability
is (for optical purposes) the same in all directions, and pro-
vided the plane of polarization is identified with the plane
which contains the magnetic vector. It is readily shown that
the direction of the ray is at right angles to the magnetic
vector and the electric force, and that the wave-front is the
plane of the magnetic vector and the electric displacement.f
After this Maxwell proceeded to investigate the propagation
of light in metals. The difference between metals and dielectrics,
so far as electricity is concerned, is that the former are con-
ductors ; and it was therefore natural to seek the cause of the
optical properties of metals in their ohmic conductivity. This
idea at once suggested a physical reason for the opacity of
metals — namely, that within a metal the energy of the light
vibrations is converted into Joulian heat in the same way as
the energy of ordinary electric currents.
* Cf. pp. 154 et sqq.
f In the memoir of 1864 Maxwell left open the choice between the above theory
and that which is obtained by assuming that in crystals the specific inductive
rapacity is (for optical purposes) the same in all directions, while the magnetic
permeability is aeolotropic. In the latcer case the plane of polarization must be
identified with the plane which contains the electric displacement. Nine years
later, in his Treatise (§ 794), Maxwell definitely adopted the former alternative.
U
290 Maxwell.
The equations of the electromagnetic field in the metal may
be written
curl H = 47rS,
- curl E = H,
S = i + D = KE +
where K denotes the ohmic conductivity ; whence it is seen that
the electric force satisfies the equation
=c2V2E.
This is of the same form as the corresponding equation in
the elastic-solid theory* ; and, like it, furnishes a satisfactory
general explanation of metallic reflexion. It is indeed correct
in all details, so long as the period of the disturbance is not too
short — i.e., so long as the light- waves considered belong to the
extreme infra-red region of the spectrum ; but if we attempt to
apply the theory to the case of ordinary light, we are confronted
by the difficulty which Lord Eayleigh indicated in the elastic-
solid theory,f and which attends all attempts to explain the
peculiar properties of metals by inserting a viscous term in
the equation. The difficulty is that, in order to account for the
properties of ideal silver, we must suppose the coefficient of
E negative — that is, the dielectric constant of the metal must
be negative, which would imply instability of electrical
equilibrium in the metal. The problem, as we have already
remarked,:}: was solved only when its relation to the theory of
dispersion was rightly understood.
At this time important developments were in progress in
the last-named subject. Since the time of Fresnel, theories of
dispersion had proceeded! from the assumption that the radii
of action of the particles of luminiferous media are so large
as to be comparable with the wave-length of light. It was
generally supposed that the aether is loaded by the molecules
* Cf. p. iso.
t Cf. p. 181. Cf. also Rayleigh, Phil. Mag. (5) xii (1881), p. 81, and
H. A. Lorentz, Over de Theorie de Terugkaatsing, Arnhem, 1875.
+ Cf. p. 181. § Cf. p. 182.
Maxwell. 291
of ponderable matter, and that the amount of dispersion
depends on the ratio of the wave-length to the distance
between adjacent molecules. This hypothesis was, however,
seen to be inadequate, when, in 1862, F. P. Leroux* found that
a prism filled with the vapour of iodine refracted the red rays
to a greater degree than the blue rays; for in all theories
which depend on the assumption of a coarse-grained lumini-
f erous medium, the refractive index increases with the frequency
of the light.
Leroux's phenomenon, to which the name anomalous dis-
persion was given, was shown by later investigators-)- to be
generally associated with " surface-colour." i.e., the property of
brilliantly reflecting incident light of some particular frequency.
Such an association seemed to indicate that the dispersive
property of a substance is intimately connected with a certain
frequency of vibration which is peculiar to that substance, and
which, when it happens to fall within the limits of the visible
spectrum, is apparent in the surface-colour. This idea of a
frequency of vibration peculiar to each kind of ponderable
matter is found in the writings of Stokes as far back as the
year 1852 ;£ when, discussing fluorescence, he remarked: —
" Nothing seems more natural than to suppose that the incident
vibrations of the luminiferous aether produce vibratory move-
ments among the ultimate molecules of sensitive substances,
and that the molecules in turn, swinging on their own account,
produce vibrations in the luminiferous aether, and thus cause
the sensation of light. The periodic times of these vibrations
depend on the periods in which the molecules are disposed to
swing, not upon the periodic time of the incident vibrations."
The principle here introduced, of considering the molecules
as dynamical systems which possess natural free periods, and
which interact with the incident vibrations, lies at the basis of
* Comptes Rendus, Iv (1862), p. 126. In 1870 C. Christiansen (Ann. d. Phys.
cxli, p. 479 ; cxliii, p. 250) observed a similar effect in a solution of fuchsin.
r Especially by Kundt, in a series of papers in the Annalen d. Phys., from
vol. cxlii (1871) onwards.
j Phil. Trans., 1852, p. 463. Stokes's Coll. Papers, iii., p. 267.
U 2
292 Maxwell.
all modern theories of dispersion. The earliest of these was
devised by Maxwell, who, in the Cambridge Mathematical Tripos
for 1869,* published the results of the following investigation : —
A model of a dispersive medium may be constituted by
embedding systems which represent the atoms of ponderable
matter in a medium which represents the aether. We may
picture each atomj- as composed of a single massive particle
supported symmetrically by springs from the interior face of
a massless spherical shell : if the shell be fixed, the particle
will be capable of executing vibrations about the centre of the
sphere, the effect of the springs being equivalent to a force on
the particle proportional to its distance from the centre. The
atoms thus constituted may be supposed to occupy small
spherical cavities in the aether, the outer shell of each atom
being in contact with the aether at all points and partaking
of its motion. An immense number of atoms is supposed to
exist in each unit volume of the dispersive medium, so that
the medium as a whole is fine-grained.
Suppose that the potential energy of strain of free aether
per unit volume is
where »j denotes the displacement and E an elastic constant ;
so that the equation of wave-propagation in free aether is
3*1 a2,,
''a? = K&
where p denotes the aethereal density.
Then if <r denote the mass of the atomic particles in unit
volume, (TJ + £) the total displacement of an atomic particle at
the place x at time t, and <rp2£ the attractive force, it is evident
that for the compound medium the kinetic energy per unit
volume is
* Cambridge Calendar, 1869 ; republished by Lord Kayleigh, Phil. Mag. xlviii
(1899), p. 151. t This illustration is due to "W. Thomson.
Maxwell. 293
and the potential energy per unit volume is
+
The equations of motion, derived by the process usual in
dynamics, are
Consider the propagation, through the medium thus constituted,
of vibrations whose frequency is n, and whose velocity of pro-
pagation in the medium is v ; so that r\ and £ are harmonic
functions of n(t - x/v). Substituting these values in the
differential equations, we obtain
1 o oil?2
Now, p/^7 has the value 1/c2, where c denotes the velocity of
light in free aether; and c/v is the refractive index ju of the
medium for vibrations of frequency n. So the equation, which
may be written
determines the refractive index of the substance for vibrations
of any frequency n. The same formula was independently
obtained from similar considerations three years later by
W. Sellmeier *
If the oscillations are very slow, the incident light being in
the extreme infra-red part of the spectrum, n is small, and the
equation gives approximately ju2 = (p + a)jp : for such oscilla-
tions, each atomic particle and its shell move together as a
rigid body, so that the effect is the same as if the aether were
simply loaded by the masses of the atomic particles, its rigidity
remaining unaltered.
* Ann. d. Phys. oxlv (1872), pp. 399, 520 : cxlvii (1872), pp. 386, 525.
Cf. also Helmholtz, Ann. d. Phys. cliv (1875), p. 582.
294 Maxwell.
The dispersion of light within the limits of the visible
spectrum is for most substances controlled by a natural
frequency p which corresponds to a vibration beyond the violet
end of the visible spectrum : so that, n being smaller than p,
we may expand the fraction in the formula of dispersion, and
obtain the equation
(T I nz n*
fJL2 = 1 + - (1 + - + -+...
f>\ P* P*
which resembles the formula of dispersion in Cauchy's theory* ;
indeed, we may say that Cauchy's formula is the expansion of
Maxwell's formula in a series which, as it converges only when
n has values within a limited range, fails to represent the
phenomena outside that range.
The theory as given above is defective in that it becomes
meaningless when the frequency n of the incident light is
equal to the frequency p of the free vibrations of the atoms.
This defect may be remedied by supposing that the motion of
an atomic particle relative to the shell in which it is contained
is opposed by a dissipative force varying as the relative
velocity ; such a force suffices to prevent the forced vibration
from becoming indefinitely great as the period of the incident
light approaches the period of free vibration of the atoms ; its
introduction is justified by the fact that vibrations in this
part of the spectrum suffer absorption in passing through the
medium. When the incident vibration is not in the same
region of the spectrum as the free vibration, the absorption is
not of much importance, and may be neglected.
It is shown by the spectroscope that the atomic systems
which emit and absorb radiation in actual bodies possess more
than one distinct free period. The theory already given may,
however, readily be extended-)- to the case in which the atoms
have several natural frequencies of vibration ; we have only to
suppose that the external massless rigid shell is connected by
springs to an interior massive rigid shell, and that this again
* Cf. p. 183.
t This subject was developed by Lord Kelvin in the' Baltimore Lectures.
Maxwell. 295
is connected by springs to another massive shell inside it, and
so on. The corresponding extension of the equation for the
refractive index is
where p^ p2, . . . denote the frequencies of the natural periods
of vibration of the atom.
The validity of the Maxwell- Sellmeier formula of disper-
sion was strikingly confirmed by experimental researches in
the closing years of the nineteenth century. In 1897 Rubens*
showed that the formula represents closely the refractive
indices of sylvin (potassium chloride) and rock-salt, with
respect to light and radiant heat of wave-lengths between
4,240 A.U. and 223,000 A.U. The constants in the formula
being known from this comparison, it was possible to predict
the dispersion for radiations of still lower frequency ; and it
was found that the square of the refractive index should have
a negative value (indicating complete reflexion) for wave-
lengths 370,000 A.U. to 550,000 A.U. in the case of rock-salt,
and for wave-lengths 450,000 ^to 670,000 A.U. in the case of
sylvin. This inference was verified experimentally in the
following year.f
It may seem strange that Maxwell, having successfully
employed his electromagnetic theory to explain the propagation
of light in isotropic media, in crystals, and in metals, should
have omitted to apply it to the problem of reflexion and refrac-
tion. This is all the more surprising, as the study of the optics
of crystals had already revealed a close analogy between the
electromagnetic theory and MacCullagh's elastic-solid theory;
and in order to explain reflexion and refraction electro-
magnetically, nothing more was necessary than to transcribe
MacCullagh's investigation of the same problem, interpreting e
(the time-flux of the displacement of MacCullagh's aether) as
the magnetic force, and curl e as the electric displacement. As
* Ann. d. Phys. Ix (1897), p. 454.
t Rubens and Aschkinass, Ann. d. Phys. Ixiv (1898).
296 Maxwell.
in MacCullagh's theory the difference between the contiguous
media is represented by a difference of their elastic constants,
so in the electromagnetic theory it may be represented by a
difference in their specific inductive capacities. From a letter
which Maxwell wrote to Stokes in 1864, and which has been
preserved,* it appears that the problem of reflexion and refrac-
tion was engaging Maxwell's attention at the time when he was
preparing his Eoyal Society memoir on the electromagnetic
field; but he was not able to satisfy himself regarding the
conditions which should be satisfied at the interface between
the media. He seems to have been in doubt which of the rival
elastic-solid theories to take as a pattern ; and it is not unlikely
that he was led astray by relying too much on the analogy
between the electric displacement and an elastic displacement. t
For in the elastic-solid theory all three components of the dis-
placement must be continuous across the interface between two
contiguous media ; but Maxwell found that it was impossible to
explain reflexion and refraction if all three components of the
electric displacement were supposed to be continuous across the
interface ; and, unwilling to give up the analogy which had
hitherto guided him aright, yet unable to disprove^ the Greenian
conditions at bounding surfaces, he seems to have laid aside the
problem until some new light should dawn upon it.
This was not the only difficulty which beset the electro-
magnetic theory. The theoretical conclusion, that the specific
inductive capacity of a medium should be equal to the square of
its refractive index with respect to waves of long period, was
not as yet substantiated by experiment; and the theory of
displacement-currents, on which everything else depended, was
* Stokes's Scientific Correspondence, ii, pp. 25, 26.
t It must be remembered tbat Maxwell pictured tbe electric displacement as a
real displacement of a medium. "My theory of electrical forces," he \vrote, " is
that they are called into play in insulating media by slight electric displacements,
which put certain small portions of the medium into a state of distortion, which,
being resisted by the elasticity of the medium, produces an electromotive force."
Campbell and Garnett's Life of Maxwell, p. 244.
| The letter to Stokes already mentioned appears to indicate that Maxwell for
a time doubted the correctness of Green's conditions.
Maxwell. 297
unfavourably received by the most distinguished of Maxwell's
contemporaries. Helmholtz indeed ultimately accepted it, but
only after many years ; and W. Thomson (Kelvin) seems never
to have thoroughly believed it to the end of his long life. In
1888 he referred to it as a "curious and ingenious, but not
wholly tenable hypothesis,"* and proposedf to replace it by an
extension of the older potential theories. In 1896 he had some
inclination? to speculate that alterations of electrostatic force
due to rapidly-changing electrification are propagated by con-
densational waves in the luminiferous aether. In 1904 he
admittedg that a bar-magnet rotating about an axis at right
angles to its length is equivalent to a lamp emitting light of
period equal to the period of the rotation, but gave his final
judgment in the sentence|| : — " The so-called electromagnetic
theory of light has not helped us hitherto."
Thomson appears to have based his ideas of the propagation
of electric disturbance on the case which had first become
familiar to him — that of the transmission of signals along a
wire. He clung to the older view that in such a disturbance
the wire is the actual medium of transmission ; whereas in
\ Maxwell's theory the function of the wire is merely to guide
the disturbance, which is resident in the surrounding dielectric.
This opinion that conductors are the media of propagation
of electric disturbance was entertained also by Ludwig Lorenz
(&. 1829, d. 1891), of Copenhagen, who independently developed
an electromagnetic theory of lightH a few years after the
publication of Maxwell's memoirs. The procedure which
Lorenz followed was that which Kiemann had suggested** in
1858 — namely, to modify the accepted formulae of electro-
dynamics by introducing terms which, though too small to be
* Nature, xxxviii (1888) p. 571. t Brit. Assoc. Report, 1888, p. 567.
J Cf. Bottouiley, in Nature, liii (1896), p. 268 ; Kelvin, ib., p. 316 ; J. Willard
Gibbs, ib., p. 509.
§ Baltimore Lectures (ed. 1904), p. 376. || Ibid., preface, p. 7.
H Oversiyt over det K. danske Vid. Selskaps Forhandliiiger, 1867, p. 26; Annul,
der Phys. cxxxi (1867), p. 243 ; Phil. Mag., xxxiv (1867), p. 287.
** Cf. p. 268. Riemann's memoir was, however, published only in the same
year (1867) as Lorenz's.
298 Maxwell.
appreciable in ordinary laboratory experiments, would be
capable of accounting for the propagation of electrical effects
through space with a finite velocity. We have seen that in
Neumann's theory the electric force E was determined by the
equation
-a, (1)
where <£ denotes the electrostatic potential defined by the
equation
4>-{\\(p'lr) dx'dy'dz',
p being the density of electric charge at the point (x, y, z'), and
where a denotes the vector-potential, defined by the equation
a={\[(i'lr)dx'dy'dz,
J J J
i' being the conduction-current at (x', y\ z'). We suppose the
specific inductive capacity and the magnetic permeability to
be everywhere unity.
Lorenz proposed to replace these by the equations
= \\\{p(t-r/c)/r\dx'dy'dz',
{i'(t-r/c)/r}dx'dy'd3f'9
the change consists in replacing the values which p and i' have
at the instant t by those which they have at the instant (t - r/c],
which is the instant at which a disturbance travelling with
velocity c must leave the place (x', y, z) in order to arrive at
the place (x, y, z) at the instant t. Thus the values of the
potentials at (x, y, z] at any instant t would, according to
Lorenz's theory, depend on the electric state at the point
(x', y', z') at the previous instant (t - r/c) : as if the potentials
were propagated outwards from the charges and currents with
velocity c. The functions <f> and a formed in this way are
generally known as the retarded potentials.
Maxwell. 299
The equations by which (f> and a have been defined are
equivalent to the equations
V2</> - $1* = - 4^, (2)
V2a - a/c2 = - 47ri, (3)
while the equation of conservation of electricity,
div i + p = 0
gives
div a + <f> = 0. (4)
From equations (1), (2), (4), we may readily derive the equation
divE = 47rcV; (I)
and from (1), (3), (4), we have
curl H = E/c2 + 47rt, (II)
where H or curl a denotes the magnetic force : while from (1)
we have
curl E = - H. (Ill)
The equations (I), (II), (III) are, however, the fundamental
equations of Maxwell's theory; and therefore the theory of
L. Lorenz is practically equivalent to that of Maxwell, so far
as concerns the propagation of electromagnetic disturbances
through free aether. Lorenz himself, however, does not appear
to have clearly perceived this ; for in his memoir he postulated
the presence of conducting matter throughout space, and was
consequently led to equations resembling those which Maxwell
had given for the propagation of light in metals. Observing
that his equations represented periodic electric currents at
right angles to the direction of propagation of the disturbance,
he suggested that all luminous vibrations might be constituted
by electric currents, and hence that there was " no longer any
reason for maintaining the hypothesis of an aether, since we
can admit that space contains sufficient ponderable matter to
enable the disturbance to be propagated."
Lorenz was unable to derive from his equations any explana-
tion of the existence of refractive indices, and his theory lacks
300 Maxwell.
the rich physical suggestiveness of Maxwell's ; the value of
his memoir lies chiefly in the introduction of the retarded
potentials. It may be remarked in passing that Lorenz's
retarded potentials are not identical with Maxwell's scalar
and vector potentials ; for Lorenz's a is not a circuital vector,
and Lorenz's <£ is not, like Maxwell's, the electrostatic potential,
but depends on the positions occupied by the charges at certain
previous instants.
For some years no progress was made either with Maxwell's
theory or with Lorenz's. Meanwhile, Maxwell had in 1865
resigned his chair at King's College, and had retired to his
estate in Dumfriesshire, where he occupied himself in writing
a connected account of electrical theory. In 1871 he returned
to Cambridge as Professor of Experimental Physics; and two
years later published his Treatise on Electricity and Magnetism.
In this celebrated work is comprehended almost every
branch of electric and magnetic theory; but the intention of
the writer was to discuss the whole as far as possible from a
single point of view, namely, that of Faraday; so that little
or no account was given of the hypotheses which had been pro-
pounded in the two preceding decades by the great German
electricians. So far as Maxwell's purpose was to disseminate
the ideas of Faraday, it was undoubtedly fulfilled ; but the
Treatise was less successful when considered as the exposition
of its author's own views. The doctrines peculiar to Maxwell
— the existence of displacement-currents, and of electromagnetic
vibrations identical with light — were not introduced in the first
'volume, or in the first half of the second volume ; and the
account which was given of them was scarcely more complete,
and was perhaps less attractive, than that which had been
furnished in the original memoirs.
Some matters were, however, discussed more fully in the
Treatise than in Maxwell's previous writings ; and among these
was the question of stress in the electromagnetic field.
It will be remembered* that Faraday, when studying the
* Cf. p. 209.
Maxwell. 301
curvature of lines of force in electrostatic fields, had noticed
an apparent tendency of adjacent lines to repel each other, as
if each tube of force were inherently disposed to distend
laterally ; and that in addition to this repellent or diverging
force in the transverse direction, he supposed an attractive or
contractile force to be exerted at right angles to it, that is to
say, in the direction of the lines of force.
Of the existence of these pressures and tensions Maxwell
was fully persuaded ; and he determined analytical expressions
suitable to represent them. The tension along the lines of
force must be supposed to maintain the ponderomotive force
which acts on the conductor on which the lines of force
terminate ; and it may therefore be measured by the force
which is exerted on unit area of the conductor, i.e., *E2/87rc2 or
iDE. The pressure at right angles to the lines of force must
then be determined so as to satisfy the condition that the aether
is to be in equilibrium.
For this purpose, consider a thin shell of aether included
between two equipotential surfaces. The equilibrium of the,
portion of this shell which is intercepted by a tube of force-
requires (as in the theory of the equilibrium of liquid films),
that the resultant force per unit area due to the above-
mentioned normal tensions on its two faces shall have the-
value T(l/pi + l//o2), where pi and pz denote the principal radii
of curvature of the shell at the place, and where T denotes,
the lateral stress across unit length of the surface of the shell,,
T being analogous to the surface-tension of a liquid film.
Now, if t denote the thickness of the shell, the area inter-
cepted on the second face by the tube of force bears to the
area intercepted on the first face the ratio (pi + t) (pz + t)/p!p2 >
and by the fundamental property of tubes of force, D and E
vary inversely as the cross-section of the tube, so the total force
on the second face will bear to that on the first face the ratio
piptKpi + 1} (pz + 1),
or approximately
302 Maxwell.
the resultant force per unit area along the outward normal is
therefore
- IDE . t . (l//t>i + I//*),
and so we have
T = - IDE . t ;
or the pressure at right angles to the lines of force is |DE per
unit area — that is, it is numerically equal to the tension along
the lines of force.
The principal stresses in the medium being thus determined,
it readily follows that the stress across any plane, to which the
unit vector N is normal, is
(D.N)E-i(D-E)N-
Maxwell obtained* a similar formula for the case of magnetic
fields ; the ponderornotive forces on magnetized matter and on
conductors carrying currents may be accounted for by assuming
a stress in the medium, the stress across the plane N" being
represented by the vector
1(B.K).H-1(B.H).N. ;j
This, like the corresponding electrostatic formula, represents a
tension across planes perpendicular to the lines of force, and a
pressure across planes parallel to them.
It may be remarked that Maxwell made no distinction
between stress in the material dielectric and stress in the
aether : indeed, so long as it was supposed that material bodies
when displaced carry the contained aether along with them,
no distinction was possible. In the modifications of Maxwell's
theory which were developed many years afterwards by his
followers, stresses corresponding to those introduced by Maxwell
were assigned to the aether, as distinct from ponderable matter ;
and it was assumed that the only stresses set up in material
bodies by the electromagnetic field are produced indirectly:
they may be calculated by the methods of the theory of
elasticity, from a knowledge of the ponderomotive forces
exerted on the electric charges connected with the bodies.
* Maxwell's Treatise on Electricity and Magnetism, § 643.
Maxwell. 303
Another remark suggested by Maxwell's theory of stress
in the medium is that he considered the question from the
purely statical point of view. He determined the stress so that
it might produce the required forces on ponderable bodies, and
be self-equilibrating in free aether. But* if the electric and
magnetic phenomena are not really statical, but are kinetic in
their nature, the stress or pressure need not be self-equilibrating.
This may be illustrated by reference to the hydrodynamical
models of the aether shortly to be described, in which perforated
solids are immersed in a moving liquid : the ponderomotive
forces exerted on the solids by the liquid correspond to those
which act on conductors carrying currents in a magnetic field,
and yet there is no stress in the medium beyond the pressure
of the liquid.
Among the problems to which Maxwell applied his theory
of stress in the medium was one which had engaged the
attention of many generations of his predecessors. The ad-
herents of the corpuscular theory of light in the eighteenth
century believed that their hypothesis would be decisively con-
firmed if it could be shown that rays of light possess momentum :
to determine the matter, several investigators directed powerful
beams of light on delicately-suspended bodies, and looked for
evidences of a pressure due to the impulse of the corpuscles.
Such an experiment was performed in 1708 by Homberg,f who
imagined that he actually obtained the effect in question ; but
Mairan and Du Fay in the middle of the century, having
repeated his operations, failed to confirm his conclusion.*
The subject was afterwards taken up by Michell, who "some
years ago," wrote Priestley § in 1772, " endeavoured to ascertain
the momentum of light in a much more accurate manner than
those in which M. Homberg and M. Mairan had attempted it."
He exposed a very thin and delicately-suspended copper plate
* Cf. V. Bjeiknes, Phil. Mag. ix (1905), p. 491.
t Histoire de 1'Acad., 1708, p. 21.
% J. J. (ie Mairan, Traite de V A urore boreale, p. 370.
§ History of Vision, i, p. 387.
304 Maxwell.
to the rays of the sun concentrated by a mirror, and observed
a deflexion. He was not satisfied that the effect of the heating
of the air had been altogether excluded, but " there seems to
be no doubt," in Priestley's opinion, " but that the motion above
mentioned is to be ascribed to the impulse of the rays of light."
A similar experiment was made by A. Bennet,* who directed
the light from the focus of a large lens on writing-paper
delicately suspended in an exhausted receiver, but " could not
perceive any motion distinguishable from the effects of heat."
" Perhaps," he concluded, " sensible heat and light may not be
caused by the influx or rectilineal projections of fine particles,
but by the vibrations made in the universally diffused caloric
or matter of heat, or fluid of light." Thus Bennet, and after
him Young, f regarded the non-appearance of light- repulsion in
this experiment as an argument in favour of the undulatory
system of light. " For," wrote Young, " granting the utmost
imaginable subtility of the corpuscles of light, their effects
might naturally be expected to bear some proportion to the
effects of the much less rapid motions of the electrical fluid,
which are so very easily perceptible, even in their weakest
states."
This attitude is all the more remarkable, because Euler
many years before had expressed the opinion that light-pressure
might be expected just as reasonably on the undulatory 'as on
the corpuscular hypothesis. "Just as," he wrote, J "a vehement
sound excites not only a vibratory motion in the particles of
the air, but there is also observed a real movement of the small
particles of dust which are suspended therein, it is not to be
doubted but that the vibratory motion set up by the light
causes a similar effect." Euler not only inferred the existence
of light-pressure, but even (adopting a suggestion of Kepler's)
accounted for the tails of comets by supposing that the solar
rays, impinging on the atmosphere of a comet, drive off from
it the more subtle of its particles.
* Phil. Trans., 1792, p. 81. + Ibid., 1802, p. 46.
J Histoire de /' Acad. de Berlin, ii (1748), p. 117.
Maxwell. 305
The question was examined by Maxwell* from the point
of view of the electromagnetic theory of light ; which readily
furnishes reasons for the existence of light-pressure. For
suppose that light falls on a metallic reflecting surface at
perpendicular incidence. The light may be regarded as con-
stituted of a rapidly-alternating magnetic field ; and this must
induce electric currents in the surface layers of the metal. But
a metal carrying currents in a magnetic field is acted on by a
ponderomotive force, which is at right angles to both the
magnetic force and the direction of the current, and is there-
fore, in the present case, normal to the reflecting surface :
this ponderomotive force is the light-pressure. Thus, according
to Maxwell's theory, light-pressure is only an extended case of
effects which may readily be produced in the laboratory.
The magnitude of the light-pressure was deduced by
Maxwell from his theory of stresses in the medium. We have
seen that the stress across a plane whose unit-normal is N is
represented by the vector
(D . N) . E - J (D . E) . N + — (B . N) . H - ~ (B . H) . N.
47T O7T
Now, suppose that a plane wave is incident perpendicularly on
a perfectly reflecting metallic sheet: this sheet must support
the mechanical stress which exists at its boundary in the
aether. Owing to the presence of the reflected wave, D is zero
at the surface ; and B is perpendicular to N, so (B . N) vanishes.
Thus the stress is a pressure of magnitude (l/8?r) (B . H)
normal to the surface : that is, the light-pressure is equal to
the density of the aethereal energy in the region immediately
outside the metal. This was Maxwell's result.
This conclusion has been reached on the assumption that
the light is incident normally to the reflecting surface. If, on
the other hand, the surface is placed in an enclosure completely
surrounded by a radiating shell, so that radiation falls on it
from all directions, it may be shown that the light-pressure is
measured by one-third of the density of aethereal energy.
* Maxwell's Treatise on Electricity and Magnetism, § 792.
X
306 Maxwell.
A different way of inferring the necessity for light-pressure
was indicated in 1876 by A. Bartoli,* who showed that, when
radiant energy is transported from a cold body to a hot one by
means of a moving mirror, the second law of thermodynamics
would be violated unless a pressure were exerted on the mirror
by the light.
The thermodynamical ideas introduced into the subject by
Bartoli have proved very fruitful. If a hollow vessel be at a
definite temperature, the aether within the vessel must be full
of radiation crossing from one side to the other : and hence the
aether, when in radiative equilibrium with matter at a given
temperature, is the seat of a definite quantity of energy per
unit volume.
If U denote this energy per unit volume, and P the light-
pressure on unit area of a surface exposed to the radiation, we
may applyf the equation of available energy!
U-TdF P
~ 1 dT
Since, as we have seen,
this equation gives .„ dU
dT'
and therefore U must be proportional to T*. From this it may
be inferred that the intensity of emission of radiant energy by
a body at temperature T is proportional to the fourth power of
the absolute temperature — a law which was first discovered
experimentally by Stefan§ in 1879.
In the year in which Maxwell's treatise was published,
Sir William Crookes|| obtained experimental evidence of a
pressure accompanying the incidence of light; but this was
* Bartoli, Sopra i movimenti prodotti dalla luce e dal calore e sopra il radiometro
di Crookes. Firenze, 1876. Also Nuovo Cimento (3) xv (1884), p. 193 ; and
Exner's Rep., xxi (1885), p. 198.
t Boltzmann, Ann. d. Phys. xxii (1884), p. 31. Cf. also B. Galitzine, Ann. d.
Phys. xlvii (1892), p. 479.
| Cf. p. 240. § Wien. Ber. Ixxix (1879), p. 391.
|| Phil. Trans, clxiv (1874), p. 501. The radiometer was discovered in 1875.
Maxwell. 307
soon found to be due to thermal effects ; and the existence of
a true light-pressure was not confirmed experimentally* until
1899. Since then the subject has been considerably developed,
especially in regard to the part played by the pressure of radiation
in cosmical physics.
Another matter which received attention in Maxwell's
Treatise was the influence of a magnetic field on the propagation
of light in material substances. We have already seenf that
the theory of magnetic vortices had its origin in Thomson's
speculations on this phenomenon ; and Maxwell in his memoir
of 1861-2 had attempted by the help of that theory to arrive
at some explanation of it. The more complete investigation
which is given in the Treatise is based on the same general
assumptions, namely, that in a medium subjected to a magnetic
field there exist concealed vortical motions, the axes of the
vortices being in the direction of the lines of magnetic force ;
and that waves of light passing through the medium disturb
the vortices, which thereupon react dynamically on the luminous
motion, and so affect its velocity of propagation.
The manner of this dynamical interaction must now be
more closely examined. Maxwell supposed that the magnetic
vortices are affected by the light-waves in the same way as
vortex-filaments in a liquid would be affected by any other
coexisting motion in the liquid. The latter problem had been
already discussed in Helrnholtz'js great memoir on vortex-
motion ; adopting Helmholtz's results, Maxwell assumed for the
additional term introduced into the magnetic force by the dis-
placement of the vortices the value 9e/B0, where e denotes the
displacement of the medium (i.e. the light vector), and the
operator d/dO denotes H^/dx + Hy'dj^y + Hzd/dz, H denoting the
imposed magnetic field. Thus the luminous motion, by dis-
turbing the vortices, gives rise to an electric current in the
medium, proportional to curl
*P. Lebedew, Archives des Sciences Phys. et Nat. (4) viii (1899), p. 184.
Ann. d. Phys. vi (1901), p. 433. E. F. Nichols and G. F. Hull, Phys. Rev.
xiii (1901), p. 293 ; Astrophys. Jour., xvii (1903), p. 315. t Cf. p. 274.
X 2
308 Maxwell.
Maxwell further assumed that the current thus produced
interacts dynamically with the luminous motion in such a
manner that the kinetic energy of the medium contains a
term proportional to the scalar product of e and curl de/30.
The total kinetic energy of the medium may therefore be
written
\p& + Jcr (e . curl 9e/a0),
where p denotes the density of the medium, and cr denotes a
constant which measures the capacity of the medium to rotate
the plane of polarization of light in a magnetic field.
The equation of motion may now be derived as in the
elastic- solid theories of light : it is
32
pe = %V2e - o- r— - curl e.
ot cu
When the light is transmitted in the direction of the lines
of force, and the axis of x is taken parallel to this direction,
the equation reduces to
and these equations, as we have seen,* furnish an explanation
of Faraday's phenomenon.
It may be remarked that the term
J(T (e . curl 9e/80)
in the kinetic energy may by partial integration be transformed
into a term
Jo- (curie. 9e/90),t
together with surface-terms ; or, again, into
- Jo- (curl e . 8e/80),
together with surf ace- terms. These different forms all yield
* Cf. p. 215.
f This form was suggested by Fitz Gerald six years later, Phil. Trans., 1880,
p. 691 : Fitz Gerald's Scientific Writings, p. 45.
Maxwell. 309
the same equation of motion for the medium; but, owing to
the differences in the surface-terms, they yield different con-
ditions at the boundary of the medium, and consequently give
rise to different theories of reflexion.
The assumptions involved in Maxwell's treatment of the
magnetic rotation of light were such as might scarcely be
justified in themselves ; but since the discussion as a whole
proceeded from sound dynamical principles, and its conclu-
sions were in harmony with experimental results, it was fitted
to lead to the more perfect explanations which were afterwards
devised by his successors. At the time of Maxwell's death,
which happened in 1879, before he had completed his forty-
ninth year, much yet remained to be done both in this and in
the other investigations with which his name is associated;
and the energies of the next generation were largely spent in
extending and refining that conception of electrical and optical
phenomena whose origin is correctly indicated in its name of
Maxwell's Theory.
( 310 )
CHAPTEK IX.
MODELS OF THE AETHER.
THE early attempts of Thomson and Maxwell to represent the
electric medium by mechanical models opened up a new field of
research, to which investigators were attracted as much by its
intrinsic fascination as by the importance of the services which
it promised to render to electric theory.
Of the models to which reference has already been made,
some — such as those described in Thomson's memoir* of 1847
and Maxwell's memoirf of 1861-2 — attribute a linear character
to electric force and electric current, and a rotatory character
to magnetism; others — such as that devised by Maxwell in
1855J and afterwards amplified by Helmholtz§ — regard mag-
netic force as a linear and electric current as a rotatory
phenomenon. This distinction furnishes a natural classification
of models into two principal groups.
Even within the limits of the former group diversity has
already become apparent ; for in Maxwell's analogy of 1861-2,
a continuous vortical motion is supposed to be in progress about
the lines of magnetic induction ; whereas in Thomson's analogy
the vector-potential was likened to the displacement in an
elastic solid, so that the magnetic induction at any point would
be represented by the twist of an element of volume of the
solid from its equilibrium position ; or, in symbols,
a = e, E = - e, B = curl e,
where a denotes the vector-potential, E the electric force, B the
magnetic induction, and e the elastic displacement.
Thomson's original memoir concluded with a notice of his
intention to resume the discussion in another communication
His purpose was fulfilled only in 1890, when|| he showed tha
* Cf. p. 270. t Of. p. 276. % Cf. p. 271. § Cf. p. 274.
|| Kelvin's Math, and Phys. Papers, iii, p. 436.
Models of the Aether. 311
in his model a linear current could be represented by a piece
of endless, cord, of the same quality as the solid and embedded
in it, if a tangential force were applied to the cord uniformly
all round the circuit. The forces so applied tangentially pro-
duce a tangential drag on the surrounding solid ; and the
rotatory displacement thus caused is everywhere proportional
to the magnetic vector.
In order to represent the effect of varying permeability,
Thomson abandoned the ordinary type of elastic solid, and
replaced it by an aether of Mac Cullagh^s type; that is to say,
an ideal incompressible substance, having no rigidity of the
ordinary kind (i.e. elastic resistance to change of shape), but
capable of resisting absolute rotation — a property to which the
name gyrostatic rigidity was given. The rotation of the solid
representing the magnetic induction, and the coefficient of
gyrostatic rigidity being inversely proportional to the permea-
bility, the normal component of magnetic induction will be
continuous across an interface, as it should be.*
We have seen above that in models of this kind the electric
force is represented by the translatory velocity of the medium.
It might therefore be expected that a strong electric field would
perceptibly affect the velocity of propagation of light ; and that
this does not appear to be the case,f is an argument against the
validity of the scheme.
We now turn to the alternative conception, in which electric
phenomena are regarded as rotatory, and magnetic force is
represented by the linear velocity of the medium; in symbols,
4-TrD = curl e,
H = e,
where D denotes the electric displacement, H the magnetic
force, and e the displacement of the medium. In Maxwell's
memoir of 1855, and in most of the succeeding writings for
* Thomson inclined to believe (Papers, iii, p. I&5) that light might he correctly
represented by the vibratory motion of such a solid.
t Wilberforce, Trans. Camb. Phil. Soc. xiv (1887), p. 170 ; Lodge, Phil. Trans,
clxxxix (1897), p. 149.
312 Models of the Aether.
many years, attention was directed chiefly to magnetic fields of
a steady, or at any rate non-oscillatory, character ; in such fields,
the motion of the particles of the medium is continuously
progressive ; and it was consequently natural to suppose the
medium to be fluid.
Maxwell himself, as we have seen,* afterwards abandoned
this conception in favour of that which represents magnetic
phenomena as rotatory. "According to Ampere and all his
followers," he wrote in 1870,f " electric currents are regarded
as a species of translation, and magnetic force as depending on
rotation. I am constrained to agree with this view, because
the electric current is associated with electrolysis, and other
undoubted instances of translation, while magnetism is asso-
ciated with the rotation of the plane of polarization of light."
But the other analogy was felt to be too valuable to be
altogether discarded, especially when in 1858 Helmholtz
extended itj by showing that if magnetic induction is com-
pared to fluid velocity, then electric currents correspond to
vortex-filaments in the fluid. Two years afterwards Kirchhoff §
developed it further. If the analogy has any dynamical (as
distinguished from a merely kinematical) value, it is evident that
the ponderomotive forces between metallic rings carrying electric
currents should be similar to the ponderomotive forces between
the same rings when they are immersed in an infinite incom-
pressible fluid; the motion of the fluid being such that its
circulation through the aperture of each ring is proportional to
the strength of the electric current in the corresponding ring.
In order to decide the question, Kirchhoff attempted, and solved,
the hydrodynamical problem of the motion of two thin, rigid
rings in an incompressible frictionless fluid, the fluid motion
being irrotational ; and found that the forces between the rings
are numerically equal to those which the rings would exert on
* Cf. p. 276.
t Proc. Lond. Math. Soc. iii (1870), p. 224 ; Maxwell's Sclent. Papers, ii, p. 263.
J Cf. p. 274.
§ Journnl fur Math. Ixxi (1869) ; Kirchhoff's Ge*amm. AbhandL, p. 404. Cf.
also C. Neumann, Leipzig Berichte, xliv (1892), p. 86.
Models of the Aether. 3 13
each other if they were traversed by electric currents pro-
portional to the circulations.
There is, however, an important difference between the two
cases, which was subsequently discussed by W. Thomson, who
pursued the analogy in several memoirs.* In order to represent
the magnetic field by a conservative dynamical system, we shall
suppose that it is produced by a number of rings of perfectly
conducting material, in which electric currents are circulating ;
the surrounding medium being free aether. Now any perfectly
conducting body acts as an impenetrable barrier to lines of
magnetic force ; for, as Maxwell showed,f when a perfect con-
ductor is placed in a magnetic field, electric currents are induced
on its surface in such a way as to make the total magnetic force
zero throughout the interior of the conductor.^ Lines of force
are thus deflected by the body in the same way as the lines of
flow of an incompressible fluid would be deflected by an obstacle
of the same form, or as the lines of flow of electric current in a
uniform conducting mass would be deflected by the introduction
of a body of this form and of infinite resistance. If, then, for
simplicity we consider two perfectly conducting rings carrying
currents, those lines of force which are initially linked with a
ring cannot escape from their entanglement, and new lines
cannot become involved in it. This implies that the total
number of lines of magnetic force which pass through the
aperture of each ring is invariable. If the coefficients of self
and mutual induction of the rings are denoted by Z,, Z2, Z12,
the electrokinetic energy of the system may be represented by
T = J (Z,*V + 2Z12^ + Z2 v),
where i\, i> denote the strengths of the currents; and the
condition that the number of lines of force linked with each
circuit is to be invariable gives the equations
Liii + Z12i2 = constant,
Lziz = constant.
* Thomson's Reprint of Papers in Elect, and Mag., §§ 573, 733, 751 (1870-
1872). t Maxwell's Treatise on Elect, and Mag., § 654.
% For this reason "W. Thomson called a perfect conductor nn ideal extreme
diamagnetic.
314 Models of the Aether.
It is evident that, when the system is considered from the
point of view of general dynamics, the electric currents must be
regarded as generalized velocities, and the quantities
(L1i1 + Z,2i2) and (Z12^ + L9i2)
as momenta. The electromagnetic ponderomotive force on the
rings tending to increase any coordinate x is dT/dv. In the
analogous hydrodynamical system, the fluid velocity corresponds
to the magnetic force: and therefore the circulation through
each ring (which is defined to be the integral fvds, taken round
a path linked once with the ring) corresponds kinematically to
the electric current ; and the flux of fluid through each ring
corresponds to the number of lines of magnetic force which
pass through the aperture of the ring. But in the hydro-
dynamical problem the circulations play the part of generalized
momenta ; while the fluxes of fluid through the rings play the
part of generalized velocities. The kinetic energy may indeed
be expressed in the form
where KI, «c2, denote the circulations (so that KI and »c2 are
proportional respectively to ^ and 4), and NI, Nn, N2, depend
on the positions of the rings ; but this is the Hamiltonian (as
opposed to the Lagrangian) form of the energy-function,* and
the ponderomotive force on the rings tending to increase
any coordinate x is - dK/dx. Since dK/dx is equal to dT/dx,
we see that the ponderomotive forces on the rings in any
position in the hydrodynamical system are equal, but opposite,
to the ponderomotive forces on the rings in the electric
system.
The reason for the difference between the two cases may
readily be understood. The rings cannot cut through the lines
of magnetic force in the one system, but they can cut through
the stream-lines in the other : consequently the flux of fluid
through the rings is not invariable when the rings are moved, the
invariants in the hydrodynamical system being the circulations.
* Cf. Whittaker, Analytical Dynamics, § 109.
Modds of the Aethtr. 315
If a thin ring, for which the circulation is zero, is introduced
into the fluid, it will experience no ponderomotive forces ; but
if a ring initially carrying no current is introduced into a
magnetic field, it will experience ponderomotive forces, owing
to the electric currents induced in it by its motion,
Imperfect though the analogy is, it is not without interest.
A bar-magnet, being equivalent to a current circulating in a wire
wound round it, may be compared (as W. Thomson remarked)
to a straight tube immersed in a perfect fluid, the fluid entering
at one end and flowing out by the other, so that the particles
of fluid follow the lines of magnetic force. If two such tubes
are presented with like ends to each other, they attract ; with
unlike ends, they repeL The forces are thus diametrically
opposite in direction to those of magnets ; but in other respects
the laws of mutual action between these tubes and between
magnets are precisely the same.*
* The mathematical analysis in this ease is very simple. A narrow rube through
which water is flowing may be regarded as equivalent to a source at one end of the
tube and a sink at the other; and the problem may therefore be reduced to the
consideration of sinks in an unlimited fluid, If there are two sinks in sneh a fluid,
of strengths m and */, the Telocity-potential is
at/r + m*//,
where r and i" denote distance from the sinks. The kinetic energy per unit
of the fluid is
thedensiryof the fluid; whence it is easily seen that the total
energy of the fluid, when the two sinks are at a dtBtance I apart, exceeds the total
cneigy when they are at an infinite distance apart by an amount
0*i*ae*i*+^ld^&m^Mmt1to+k&™l»m*at1i*>
small spheres *, /, surrounding the sinks. By Green's
reduces at once to
where the integration is taken over * and «", and m
or »'. The integral taken over *' vanishe
hare
of the fluid is therefore greater when sinks of strengths at, at* are at a
3 1 6 Models of the Aether.
Thomson, moreover, investigated* the ponderomotive forces
which act between two solid bodies immersed in a fluid, when
one of the bodies is constrained to perform small oscillations.
If, for example, a small sphere immersed in an incompressible
fluid is compelled to oscillate along the line which joins its
centre to that of a much larger sphere, which is free, the free
sphere will be attracted if it is denser than the fluid ; while
if it is less dense than the fluid, it will be repelled or attracted
according as the ratio of its distance from the vibrator to its
radius is greater or less than a certain quantity depending on
the ratio of its density to the density of the fluid. Systems
of this kind were afterwards extensively investigated by
C. A. Bjerknes.f Bjerknes showed that two spheres which
are immersed in an incompressible fluid, and which pulsate
(i.e., change in volume) regularly, exert on each other (by the
mediation of the fluid) an attraction, determined by the inverse
square law, if the pulsations are concordant ; and exert on
each other a repulsion, determined likewise by the inverse
square law, if the phases of the pulsations differ by half a
period. It is necessary to suppose that the medium is incom-
pressible, so that all pulsations are propagated instantaneously :
otherwise attractions would change to repulsions and vice versa
at distances greater than a quarter wave-length.^ If the
spheres, instead of pulsating, oscillate to and fro in straight
lines about their mean positions, the forces between them are
proportional in magnitude and the same in direction, but
mutual distance I than when sinks of the same strengths are at infinite distance
apart by an amount lirpmm'/l. Since, in the case of the tubes, the quantities m
correspond to the fluxes of fluid, this expression corresponds to the Lagrangian
form of the kinetic energy ; and therefore the force tending to increase the coordi-
nate x of one of the sinks is (3/9#) (4ny> ww'/Z). "Whence it is seen that the like ends
of two tubes attract, and the unlike ends repel, according to the inverse square la\\r.
* Phil. Mag. xli (1870), p. 427.
t Repertorium d. Mathematik von Konisberger und Zeuner (1876), p. 268.
Gottinger Nachrichten, 1876, p. 245. Comptes Rendus, Ixxxiv (1877), p. 1377.
Cf. Nature, xxiv (1881), p. 360.
J On the mathematical theory of the force between two pulsating spheres in
a fluid, cf. W. M. Hicks, Proc. Camb. Phil. Soc. iii (1879), p. 276 ; iv (1880),
p. 29.
Models of the Aether. 3 1 7
opposite in sign, to those which act between two magnets
oriented along the directions of oscillation.*
The results obtained by Bjerknes were extended by
A. H. Leahyf to the case of two spheres pulsating in an
elastic medium ; the wave-length of the disturbance being
supposed large in comparison with the distance between the
spheres. For this system Bjerknes' results are reversed, the
law being now that of attraction in the case of unlike phases,
and of repulsion in the case of like phases : the intensity is as
before proportional to the inverse square of the distance.
The same author afterwards discussed \ the oscillations
which may be produced in an elastic medium by the
displacement, in the direction of the tangent to the cross-
section, of the surfaces of tubes of small sectional area :
the tubes either forming closed curves, or extending inde-
finitely in both directions. The direction and circumstances
of the motion are in general analogous to ordinary vortex-
motions in an incompressible fluid ; and it was shown by Leahy
that, if the period of the oscillation be such that the waves
produced are long compared with ordinary finite distances, the
displacement due to the tangential disturbances is proportional
to the velocity due to vortex-rings of the same form as the
tubular surfaces. One of these " oscillatory twists," as the
tubular surfaces may be called, produces a displacement which
is analogous to the magnetic force due to a current flowing in
a curve coincident with the tube ; the strength of the current
being proportional to b'w sin pt, where b denotes the radius of
the twist, and t» sin pt its angular displacement. If the field
of vibration is explored by a rectilineal twist of the same
period as that of the vibration, the twist will experience a force
* A theory of gravitation has heen hased by Korn on the assumption that
gravitating particles resemhle slightly compressible spheres immersed in an incom-
pressible perfect fluid : the spheres execute pulsations, whose intensity corresponds
to the mass of the gravitating particles, and thus forces of the Newtonian kind are
produced between them. Cf. Korn, Eine Theorie der Gravitation und der elect.
Etscheinungen, Berlin, 1898.
t Trans. Camb. Phil. Soc. xiv (1884), p. 45.
J Trans. Camb. Phil. Soc. xiv (1885), p. 188.
3 1 8 Models of the Aether.
at right angles to the plane containing the twist and the
direction of the displacement which would exist if the twist
were removed ; if the displacement of the medium be repre-
sented by F sin pt, and the angular displacement of the twist
by w sin pt, the magnitude of the force is proportional to the
vector-product of V (in the direction of the displacement) and
w (in the direction of the axis of the twist).
A model of magnetic action may evidently be constructed
on the basis of these results. A bar-magnet must be regarded
as vibrating tangentially, the direction of vibration being
parallel to the axis of the body. A cylindrical body carrying
a current will have its surface also vibrating tangentially ; but
in this case the direction of vibration will be perpendicular to
the axis of the cylinder. A statically electrified body, on the
other hand, may, as follows from the same author's earlier work,
be regarded as analogous to a body whose surface vibrates in
the normal direction.
We have now discussed models in which the magnetic force
is represented as the velocity in a liquid, and others in which
it is represented as the displacement in an elastic solid. Some
years before the date of Leahy's memoir, George Francis
Fitz Gerald (b. 1851, d. 1901)* had instituted a comparison
between magnetic force and the velocity in a quasi-elastic
solid of the type first devised by MacCullagh.f An analogy
is at once evident when it is noticed that the electromagnetic
equation
4?rD = curl H
is satisfied identically by the values
4?rD = curl e,
H = e,
where e denotes any vector; and that, on substituting these
values in the other electromagnetic equation,
- curl (4ircsD/e) - H, •
* Phil. Trans., 1880, p. 691 (presented October, 1878). Fitz Gerald's Scientific
Writings, p. 45. t Cf. p. 155.
Moaeh of the Aether. 319
we obtain the equation
ee + c2 curl curl e = 0,
which is no other than the equation of motion of MacCullagh's
aether,* the specific inductive capacity £ corresponding to the
reciprocal of MacCullagh's constant of elasticity. In the
analogy thus constituted, electric displacement corresponds to
the twist of the elements of volume of the aether ; and electric
charge must evidently be represented as an intrinsic rotational
strain. Mechanical models of the electromagnetic field, based on
Fitz Gerald's analogy, were afterwards studied by A. Sommerfeld,f
by K. Keiff,J and by Sir J. Larmor.§ The last-named authorll
supposed the electric charge to exist in the form of discrete
electrons, for the creation of which he suggested the following
ideal processIF : — A filament of aether, terminating at two
nuclei, is supposed to be removed, and circulatory motion is
imparted to the walls of the channel so formed, at each point
of its length, so as to produce throughout the medium a
rotational strain. When this has been accomplished, the
channel is to be filled up again with aether, which is to be
made continuous with its walls. When the constraint is
removed from the walls of the channel, the circulation imposed
on them proceeds to undo itself, until this tendency is balanced
by the elastic resistance of the aether with which the channel
has been filled up ; thus finally the system assumes a state of
equilibrium in which the nuclei, which correspond to a positive
and a negative electron, are surrounded by intrinsic rotational
strain.
Models in which magnetic force is represented by the
velocity of an aether are not, however, secure from objection.
It is necessary to suppose that the aether is capable of flowing
like a perfect fluid in irrotational motion (which would corre-
* Cf. p. 155. t Ann. d. Phys. xlvi (1892), p. 139.
I Reiff, Elasticitat und Elektricitdt, Freiburg, 1893.
§ Phil. Trans, clxxxv (1893), p. 719.
|| In a supplement, of date August, 1894, to his above-cited memoir of 1893.
H Phil. Trans, clxxxv (1894), p. 810; cxc (1897), p. 210; Larmor, Aether
.and Matter (1900), p. 326.
320 Models of the Aether.
spond to a steady magnetic field), and that it is at the same
time endowed with the power (which is requisite for the
explanation of electric phenomena) of resisting the rotation of
any element of volume.* But when the aether moves irrota-
tionally in the fashion which corresponds to a steady magnetic
field, each element of volume acquires after a finite time a
rotatory displacement from its original orientation, in con-
sequence of the motion ; and it might therefore be expected that
the quasi-elastic power of resisting rotation would be called
into play — i.e., that a steady magnetic field would develop
electric phenomena.f
A further objection to all models in which magnetic force
corresponds to velocity is that a strong magnetic field, being in
such models represented by a steady drift of the aether, might
be expected to influence the velocity of propagation of light.
The existence of such an effect appears, however, to be disproved
by the experiments of Sir Oliver Lodge ; J at any rate, unless it
is assumed that the aether has an inertia at least of the same
order of magnitude as that of ponderable matter, in which case
the motion might be too slow to be measurable.
Again, the evidence in favour of the rotatory as opposed to
the linear character of magnetic phenomena has perhaps, on the
whole, been strengthened since Thomson originally based his
conclusion on the magnetic rotation of light. This brings us
to the consideration of an experimental discovery.
In 1879 E. H. Hall,§ at that time a student at Baltimore,
* Larmor (loc. cit.) suggested the analogy of a liquid filled with magnetic
molecules under the action of an external magnetic field.
It has often heen objected to the mathematical conception of a perfect fluid
that it contains no safeguard against slipping between adjacent layers, so that
there is no justification for the usual assumption that the motion of <i perfect fluid
is continuous. Larmor remarked that a rotational elasticity, such as is attributed
to the medium above considered, furnishes precisely such a safeguard ; and that
without some property of this kind a continuous frictionless fluid cannot be imagined.
t Larmor proposed to avoid this by assuming that the rotation which is resisted
by an element of volume of the aether is the vector sum of the series of differential
rotations which it has experienced. J Phil. Trans, clxxxix (1897), p. 149.
§ Am. Jour. Math, ii, p. 287 ; Am. J. Sci. xix, p. 200, and xx, p. 161 ; Phil.
Mag. ix, p. 225, and x, p. 301.
Models of the Aether. 321
repeating an experiment which had been previously suggested
by H. A. Kowland, obtained a new action of a magnetic field
on electric currents. A strip of gold leaf mounted on glass,
forming part of an electric circuit through which a current
was passing, was placed between the poles of an electro-
magnet, the plane of the strip being perpendicular to the
lines of magnetic force. The two poles of a sensitive galvano-
meter were then placed in connexion with different parts of the
strip, until two points at the same potential were found. When
the magnetic field was created or destroyed, a deflection of the
galvanometer needle was observed, indicating a change in the
relative potential of the two poles. It was thus shown that
the magnetic field produces in the strip of gold leaf a new
electromotive force, at right angles to the primary electromotive
force and to the magnetic force, and proportional to the product
of these forces.
From the physical point of view we may therefore regard
Hall's effect as an additional electromotive force generated by
the action of the magnetic field on the current ; or alternatively
we may regard it as a modification of the ohmic resistance of
the metal, such as would be produced if the molecules of the
metal assumed a helicoidal structure about the lines of magnetic
force. From the latter point of view, all that is needed is
to modify Ohm's law
S = £E
(where S denotes electric current, k specific conductivity, and E
electric force) so that it takes the form
S = KE + h [E . H]
where H denotes the imposed magnetic force, and h denotes a
constant on which the magnitude of Hall's phenomenon
depends. It is a curious circumstance that the occurrence, in
the case of magnetized bodies, of an additional term in Ohm's
law, formed from a vector-product of E, had been expressly
suggested in Maxwell's Treatise*: although Maxwell had not-
indicated the possibility of realizing it by Hall's experiment.
* Elect, and Mag., § 303. Cf. Hopkinson, Phil. Mag. x (1880), p. 430.
Y
322 Models of the Aether.
An interesting application of Hall's discovery was made in
the same year by Boltzmann,* who remarked that it offered a
prospect of determining the absolute velocity of the electric
charges which carry the current in the strip. For if it is
supposed that only one kind (vitreous or resinous) of electricity
is in motion, the force on one of the charges tending to drive it
to one side of the strip will be proportional to the vector-
product of its velocity and the magnetic intensity. Assuming
that Hall's phenomenon is a consequence of this tendency of
charges to move to one side of the strip, it is evident that the
velocity in question must be proportional to the magnitude of
the Hall electromotive force due to a unit magnetic field. On
the basis of this reasoning, A. von Ettingshausenf found for the
current sent by one or two Daniell's cells through a gold strip
a velocity of the order of 0*1 cm. per second. It is clear, however,
that, if the current consists of both vitreous and resinous charges
in motion in opposite directions, Boltzmann's argument fails ;
for the two kinds of electricity would give opposite directions
to the current in Hall's phenomenon.
In the year following his discovery, Hall} extended his
researches in another direction, by investigating whether a
magnetic field disturbs the distribution of equipotential lines in
a dielectric which is in an electric field ; but no effect could be
observed.§ Such an effect, indeed,|| was not to be expected on
theoretical grounds; for when, in a material system, all the
velocities are reversed, the motion is reversed, it being
understood that, in the application of this theorem to electrical
theory, an electrostatic state is to be regarded as one of rest, and
a current as a phenomenon of motion ; and if such a reversal be
* Wien Anz., 1880, p. 12. Phil. Mag. ix (1880), p. 307.
t Ann. d. Phys. xi (1880), pp. 432, 1044.
1 Am. Jour. Sci. xx (1880), p. 164.
§ In 1885-6 E. van Aubel, Bull, de 1'Acad. Roy. de Belgique (3) x, p. 609 ;
xii, p. 280, repeated the investigation in an improved form, and confirmed the
result that a magnetic field has no influence on the electrostatic polarization of
dielectrics.
|| H. A. Lorentz, Arch. Neerl. xix (1884), p. 123.
Models of the Aether. 323
performed in the present system, the poles of the electro-
magnet are exchanged, while in the dielectric no change takes
place.
We must now consider the bearing of Hall's effect on
the question as to whether magnetism is a rotatory or a
linear phenomenon.* If magnetism be linear, electric currents
must be rotatory; and if Hall's phenomenon be supposed to take
place in a horizontal strip of metal, the magnetic force being
directed vertically upwards, and the primary current flowing
horizontally from north to south, the only geometrical entities
involved are the vertical direction and a rotation in the east-
and-west vertical plane ; and these are indifferent with respect
to a rotation in the nor th-and- south vertical plane, so that there
is nothing in the physical circumstances of the system to
determine in which direction the secondary current shall flow.
The hypothesis that magnetism is linear appears therefore
to be inconsistent with the existence of Hall's effect, f There
are, however, some considerations which may be urged on the
other side. Hall's effect, like the magnetic rotation of light,
takes place only in ponderable bodies, not in free aether ; and
its direction is sometimes in one sense, sometimes in the other,
according to the nature of the substance. It may therefore be
doubted whether these phenomena are not of a secondary
character, and the argument based on them invalid. Moreover,
as Fitz Gerald remarked,^ the magnetic lines of force associated
with a system of currents are circuital and have no open ends,
making it difficult to imagine how alteration of rotation inside
them could be produced.
Of the various attempts to represent electric and magnetic
phenomena by the motions and strains of a continuous medium,
none of those hitherto considered has been found free from
* Of. F. Kol&cek, Ann. d. Phys. Iv (1895), p. 503.
t Further evidence in favour of the hypothesis that it is the electric phenomena
which are linear is furnished by the fact that pyro-electric effects (the production of
electric polarization by warming) occur in acentric crystals, and only in such. Cf.
M. Abraham, Encyklopiidie der rnrith. Wiss. iv (2), p. 43.
I Cf. Larmor, Phil. Trans, clxxxv, p. 780.
Y 2
324 Models of the Aether.
objection.* Before proceeding to consider models which are not
constituted by a continuous medium, mention must be made of
a suggestion offered by Biemann in his lecturesf of 1861. Rie-
mann remarked that the scalar-potential 0 and vector-potential
a, corresponding to his own law of force between electrons,
satisfy the equation
0 + div a = 0 ;
an equation which, as we have seen, is satisfied also by the
potentials of L. Lorenz.j This appeared to Riemann to indicate
that <j> might represent the density of an aether, of which a
represents the velocity. It will be observed that on this
hypothesis the electric and magnetic forces correspond to second
derivates of the displacement — a circumstance which makes it
somewhat difficult to assimilate the energy possessed by the
electromagnetic field to the energy of the model.
We must now proceed to consider those models in which
the aether is represented as composed of more than one kind of
constituent : of these Maxwell's model of 1861-2, formed of
vortices and rolling particles, may be taken as the type. Another
device of the same class was described in 1885 by Fitz Gerald§ ;
this was constituted of a number of wheels, free to rotate on
axes fixed perpendicularly in a plane board ; the axes were fixed
at the intersections of two systems of perpendicular lines ; and
each wheel was geared to each of its four neighbours by an
indiarubber band. Thus all the wheels could rotate without
any straining of the system, provided they all had the same
angular velocity; but if some of the wheels were revolving
faster than others, the indiarubber bands would become strained.
It is evident that the wheels in this model play the same part
as the vortices in Maxwell's model of 1861-2 : their rotation is
* Cf. H. "Witte, Ueber den gegenwdrtigen Stand der Frage nach einer mecha-
nischen Erkldrung der elektrischen Erscheinungen ; Berlin, 1906.
t Edited after his death by K. Hattendorff, under the title Schwere, Elektricitiit,
und Magnetismus, 1875, p. 330.
I Cf. p. 299.
§ Scient. Proc. Koy. Dublin Soc., 1885; Phil. Mag. June, 1885; Fitz Gerald's
Seient. Writings, pp. 142, 157.
^B Models of the Aether. 325
the analogue of magnetic force ; and a region in which the masses
of the wheels are large corresponds to a region of high magnetic
permeability. The indiarubber bands of Fitz Gerald's model
correspond to the medium in which Maxwell's vortices were
embedded ; and a strain on the bands represents dielectric polari-
zation, the line joining the tight and slack sides of any band
being the direction of displacement. A body whose specific
inductive capacity is large would be represented by a region
in which the elasticity of the bands is feeble. Lastly,
conduction may be represented by a slipping of the bands
on the wheels.
Such a model is capable of transmitting vibrations analogous
to those of light. For if any group of wheels be suddenly set
in rotation, those in the neighbourhood will be prevented by
their inertia from immediately sharing in the motion; but
presently the rotation will be communicated to the adjacent
wheels, which will transmit it to their neighbours; and so a
wave of motion will be propagated through the medium. The
motion constituting the wave is readily seen to be directed in
the plane of the wave, i.e. the vibration is transverse. The axes
of rotation of the wheels are at right angles to the direction
of propagation of the wave, and the direction of polarization of
the bands is at right angles to both these directions.
The elastic bands may be replaced by lines of governor
balls :* if this be done, the energy of the system is entirely of
the kinetic type.f
Models of types different from the foregoing have been
suggested by the researches of Helmholtz and W. Thomson on
vortex-motion. The earliest attempts in this direction, however,
were intended to illustrate the properties of ponderable matter
rather than of the luminiferous medium. A vortex existing in
a perfect fluid preserves its individuality throughout all changes,
* Fitz Gerald's Scient. Writings, p. 271.
t It is of course possible to devise models of this class in which the rotation may
be interpreted as having the electric instead of the magnetic character. Such a
model was proposed by Boltzinann, Vorlesungen iiber Maxwell's Theorie, ii.
326 Models of the Aetker.
and cannot be destroyed ; so that if, as Thomson* suggested in
1867, the atoms of matter are constituted of vortex-rings in a
perfect fluid, the conservation of matter may be immediately
explained. The mutual interactions of atoms may be illustrated
by the behaviour of smoke-rings, which after approaching each
other closely are observed to rebound : and the spectroscopic
properties of matter may be referred to the possession by
vortex-rings of free periods of vibration. f
There are, however, objections to the hypothesis of vortex-
atoms. It is not easy to understand how the large density of
ponderable matter as compared with aether is to be explained ;
and further, the virtual inertia of a vortex-ring increases as its
energy increases ; whereas the inertia of a ponderable body is,
so far as is known, unaffected by changes of temperature. It
is, moreover, doubtful whether vortex-atoms would be stable.
" It now seems to me certain," wrote W. Thomson^ (Kelvin) in
1905, " that if any motion be given within a finite portion of
an infinite incompressible liquid, originally at rest, its fate is
necessarily dissipation to infinite distances with infinitely small
velocities everywhere; while the total kinetic energy remains
constant. After many years of failure to prove that the motion
in the ordinary Helmholtz circular ring is stable, I came to the
conclusion that it is essentially unstable, and that its fate must
be to become dissipated as now described."
The vortex-atom hypothesis is not the only way in which
the theory of vortex-motion has been applied to the construc-
tion of models of the aether. It was shown in 1880 by
W. Thomson§ that in certain circumstances a mass of fluid can
exist in a state in which portions in rotational and irrotational
* Phil. Mag. xxxiv(1867), p. 15; Proc. R.S. Edinb. vi, p. 94.
t An attempt was made in 1883 by J. J. Thomson, Phil. Mag. xv (1883),
p. 427, to explain the phenomena of the electric discharge through gases in terms
of tho theory of vortex-atoms. The electric field was supposed to consist in a
distribution of velocity in the medium whose vortex-motion constituted the atoms
of the gas ; and Thomson considered the effect of this field on the dissociation and
recoupling of vortex-rings.
J Proc. Roy. Soc. Edinb., xxv (1905), p. 565.
§ Brit. Assoc. Rep., 1880, p. 473.
Models of the Aether. 327
motion are finely mixed together, so that on a large scale the
mass is homogeneous, having within any sensible volume an
equal amount of vortex-motion in all directions. To a fluid
having such a type of motion he gave the name vortex-sponge.
TiveTyears later, Fitzgerald*" discussed the suitability of the
vortex-sponge as a model of the aether. Since vorticity in a
perfect fluid cannot be created or destroyed, the modification
of the system which is to be analogous to an electric field must
be a polarized state of the vortex motion, and light must be
represented by a communication of this polarized motion from
one part of the medium to another. Many distinct types of
polarization may readily be imagined : for instance, if the
turbulent motion were constituted of vortex-rings, these might
be in motion parallel to definite lines or planes ; or if it were
constituted of long vortex filaments, the filaments might be
bent spirally about axes parallel to a given direction. The
energy of any polarized state of vortex-motion would be greater
than that of the unpolarized state; so that if the motion of
matter had the effect of reducing the polarization, there would
be forces tending to produce that motion. Since the forces due
to a small vortex vary inversely as a high power of the distance
from it, it seems probable that in the case of two infinite
planes, separated by a region of polarized vortex-motion, the
forces due to the polarization between the planes would depend
on the polarization, but not on the mutual distance of the
planes — a property which is characteristic of plane distributions
whose elements attract according to the Newtonian law.
It is possible to conceive polarized forms of vortex- motion
which are steady so far as the interior of the medium is
concerned, but which tend to yield up their energy in producing
motion of its boundary — a property parallel to that of the
aether, which, though itself in equilibrium, tends to move
objects immersed in it.
In the same year Hicksf discussed the possibility of trans-
* Scient. Proc. Roy. Dublin Soc., 188o • Scientific Wntings of FitzGerakl,
p. 154. + Brit. Assoc. Rep., 1885, p. 930.
328 Models of the Aether.
mitting waves through a medium consisting of an incompressible
fluid in which small vortex-rings are closely packed together.
The wave-length of the disturbance was supposed large in com-
parison with the dimensions and mutual distances of the rings ;
and the translatory motion of the latter was supposed to be so
slow that very many waves can pass over any one before it has
much changed its position. Such a medium would probably
act as a fluid for larger motions. The vibration in the wave-
front might be either swinging oscillations of a ring about a
diameter, or transverse vibrations of the ring, or apertural
vibrations ; vibrations normal to the plane of the ring appear
to be impossible. Hicks determined in each case the velocity
of translation, in terms of the radius of the rings, the distance
of their planes, and their cyclic constant.
The greatest advance in the vortex-sponge theory of the
aether was made in 1887, when W. Thomson* showed that the
equation of propagation of laminar disturbances in a vortex-
sponge is the same as the equation of propagation of luminous
vibrations in the aether. The demonstration, which in the
circumstances can scarcely be expected to be either very simple
or very rigorous, is as follows : —
Let (u, v, w) denote the components of velocity, and p the
pressure, at the point (x, y, z) in an incompressible fluid. Let
the initial motion be supposed to consist of a laminar motion
{/(?/), 0, Oj, superposed on a homogeneous, isotropic, and fine-
grained distribution (u'0t v0, w0) : so that at the origin of time
the velocity is {/ (y) + u'0, v0, wn\ : it is desired to find a
function / (y, t) such that at any time t the velocity shall
be \f(y, t) + u', v, w), where u', v, w, are quantities of which
every average taken over a sufficiently large space is zero.
Substituting these values of the components of velocity in
the equation of motion
du _ du du du dp
dt dx ~ dy~ dz ~ dx'
* Phil. Mug. xxiv (1887), p. 342 : Kelvin's Math, and Phys. Papers, iv, p. 308.
Models of the Aether. 329
there results
W dp
- w — - £.
dz dx
Take now the #2-averages of both members. The quantities
du'/dt, du'/dx, v, dp/dx have zero averages; so the equation
takes the form
df(y*t) ( ,W M
- = - A . [u -- + v — + w
dt \ dx dy
if the symbol A is used to indicate that the xz- average is to be
taken of the quantity following. Moreover, the incompressi-
bility of the fluid is expressed by the equation
whence
du' dv dw
+ ~ + = '
f\ A I / *** / t/*1' ^ \JWJ
1 aaT"1 * ^+ l 9z
When this is added to the preceding equation, the first and
third pairs of terms of the second member vanish, since the
^-average of any derivate dQ/dx vanishes if Q is finite for
infinitely great values of x ; and the equation thus becomes
a)
From this it is seen that if the turbulent motion were to remain
continually isotropic as at the beginning,/ (T/, t) would constantly
retain its critical value /(y). In order to examine the deviation
from isotropy, we shall determine Ad (u'v)/dt, which may be
done in the following way : — Multiplying the u- and ^-equations
of motion by v, u' respectively, and adding, we have
-.
' fa ty dx
d (u'v) d (u'v) dp , dp
-V- - ~ w ~V— -v^--uf^-
dy dz dx ty
330 Models of the Aether.
Taking the ^-average of this, we observe that the first term of
the first member disappears, since A . v is zero, and the first
term of the second member disappears, since A . 3 (u'v]fix is
zero. Denoting by %RZ the average value of uz, vz, or w1, so
that R may be called the average velocity of the turbulent
motion, the equation becomes
It ^ • (-
V * #2 y^
' y O
9)jm 3-"' ^
Vj
where
n. i la> *«'*>,,
9(^) d(u
y) 9p , 3p
i ^ _ f u
dx dy
Let p be written (jp' + TO), where y denotes the value which p
would have if / were zero. The equations of motion immediately
give
and on subtracting the forms which this equation takes in the
two cases, we have
which, when the turbulent motion is fine-grained, so that
f(yt t) is sensibly constant over ranges within which u't v, w
pass through all their values, may be written
Moreover, we have
. , , tyu'v) d(uv)
0
for positive and negative values of u, v, w are equally probable ;
and therefore the value of the second member of this equation
is doubled by adding to itself what it becomes when for u', v, w
we substitute - u', -v, -w, which (as may be seen by inspection
of the above equation in V2^>) does not change the value of p'.
Models of the Aether. 331
Comparing this equation with that which determines the value
of Q, we have
' d^
or substituting for CT,
The isotropy with respect to x and z gives the equation
, 8 a\ 8 _
-+ ^0- h" V ^«
But by integration by parts we obtain the equation
' U.v-^o=_
and by the condition of incompressibility the second member
may be written
A . (tojty) . (d/ty) . V-2Vo, or - A . v0 . (d/zdf) . V-^o ;
so we have
On account of the isotropy, we may write J for
and, therefore,
The deviation from isotropy shown by this equation is very
small, because of the smallness of df(y, t)/dy. The equation is
therefore not restricted to the initial values of the two members,
332 Models of the Aether.
for we may neglect an infinitesimal deviation from (2/9) IP in
the first factor of the second member, in consideration of the
smallness of the second factor. Hence for all values of t we
have the equation
which, in combination with (1), yields the result
the form of this equation shows that laminar disturbances are
propagated through the vortex-sponge in the same manner as waves
of distortion in a homogeneous elastic solid.
The question of the stability of the turbulent motion remained
undecided ; and at the time Thomson seems to have thought it
likely that the motion would suffer diffusion. But two years
later* he showed that stability was ensured at any rate when
space is filled with a set of approximately straight hollow vortex
filaments. Fitz Geraldf subsequently determined the energy per
unit-volume in a turbulent liquid which is transmitting laminar
waves. Writing for brevity
(2/9) R* - V\ f(y, t) = P, and A (u'v) = 7,
the equations are
s?.--h and h-.y*^
dt dy' ft " 8y
If the quantity
p-f jVP«2S
is integrated throughout space, and the variations of the
integral with respect to time are determined, it is found that
JIM-
* Proc. Roy. Irish Acad. (3) i (1889), p. 340 ; Kelvin's Math, and Phys. Papers,
iv, p. 202.
t Brit. Assoc. Rep., 1899. Fitz Gerald's Scientific Writings, p. 484.
Models of the Aether. 333
Integrating the second term under the integral by parts, and
omitting the superficial terms (which may be at infinity, or
wherever energy enters the space under consideration), we have
0fa***.JJJp(*+g
Hence it appears that the quantity S, which is of the dimensions
of energy, must be proportional to the energy per unit- volume
of the medium — a result which shows that there is a pronounced
similarity between the dynamics of a vortex- sponge and of
Maxwell's elastic aether.
A definite vortex-sponge model of the aether was described
by Hicks in his Presidential Address to the mathematical
section of the British Association in 1895.* In this the small
motions whose function is to confer the quasi-rigidity were not
completely chaotic, but were disposed systematically. The
medium was supposed to be constituted of cubical elements of
fluid, each containing a rotational circulation complete in itself :
in any element, the motion close to the central vertical diameter
of the element is vertically upwards : the fluid which is thus
carried to the upper part of the element flows outwards over
the top, down the sides, and up the centre again. In each of
the six adjoining elements the motion is similar to this, but in
the reverse direction. The rotational motion in the elements
confers on them the power of resisting distortion, so that waves
may be propagated through the medium as through an elastic
solid ; but the rotations are without effect on irrotational
motions of the fluid, provided the velocities in the irrotational
motion are slow compared with the velocity of propagation of
distortional vibrations.
A different model was described four years later by
Fitz Gerald, f Since the distribution of velocity of a fluid in the
* Brit. Assoc. Rep., 1895, p. 595.
t Proc. Roy. Dublin Soc., December 12, 1899; Fitz Gerald's Scientific
Writings, p. 472.
334 Models of the Aether.
neighbourhood of a vortex filament is the same as the distribu-
tion of magnetic force around a wire of identical form carrying
an electric current, it is evident that the fluid has more energy
when the filament has the form of a helix than when it is
straight ; so if space were filled with vortices, whose axes
were all parallel to a given direction, there would be an
increase in the energy per unit volume when the vortices
were bent into a spiral form ; and this could be measured by
the square of a vector — say, E — which may be supposed parallel
to this direction.
If now a single spiral vortex is surrounded by parallel
straight ones, the latter will not remain straight, but will be
bent by the action of their spiral neighbour. The transference
of spirality may be specified by a vector H, which will be dis-
tributed in circles round the spiral vortex ; its magnitude will
depend on the rate at which spirality is being lost by the
original spiral, and can be taken such that its square is equal
to the mean energy of this new motion. The vectors E and H
will then represent the electric and magnetic vectors; the
vortex spirals representing tubes of electric force.
Fitz Gerald's spirality is essentially similar to the laminar
motion investigated by Lord Kelvin, since it involves a flow in
the direction of the axis of the spiral, and such a flow cannot
take place along the direction of a vortex filament without a
spiral deformation of a filament.
Other vortex analogues have been devised for electro-
statical systems. One such, which was described in 1888 by
W. M. Hicks,* depends on the circumstance that if two bodies
in contact in an infinite fluid are separated from each other, and
if there be a vortex filament which terminates on the bodies,
there will be formed at the point where they separate a hollow
vortex filamentf stretching from one to the other, with rotation
* Brit. Assoc. Rep., 1888, p. 577.
} A hollow vortex is a cyclic motion existing in a fluid without the presence of
any actual rotational filaments. On the general theory cf. Hicks, Phil. Trans,
clxxv (1883), p. 161 ; clxxvi (1885), p. 725 ; cxcii (1898), p. 33.
Models of the Aether. 335
equal and opposite to that of the original filament. As the
bodies are moved apart, the hollow vortex may, through failure
of stability, dissociate into a number of smaller ones ; and if
the resulting number be very large, they will ultimately take
up a position of stable equilibrium. The two sets of filaments
—the original filaments and their hollow companions — will be
intermingled, and each will distribute itself according to the
same law as the lines of force between the two bodies which are
equally and oppositely electrified.
Since the pressure inside a hollow vortex is zero, the portion
of the surface on which it abuts experiences a diminution of
pressure ; the two bodies are therefore attracted. Moreover, as
the two bodies separate further, the distribution of the filaments
being the same as that of lines of electric force, the diminution
of pressure for each line is the same at all distances, and there-
fore the force between the two bodies follows the same law as
the force between two bodies equally and oppositely electrified.
It may be shown that the effect of the original filaments is
similar, the diminution of pressure being half as large again as
for the hollow vortices.
If another surface were brought into the presence of the
others, those of the filaments which encounter it would break
off and rearrange themselves so that each part of a broken
filament terminates on the new body. This analogy thus gives
a complete account of electrostatic actions both quantitatively
and qualitatively : the electric charge on a body corresponds
to the number of ends of filaments abutting on it, the sign
being determined by the direction of rotation of the filament
as viewed from the body.
A magnetic field may be supposed to be produced by the
motion of the vortex filaments through the stationary aether,
the magnetic force being at right angles to the filament and to
its direction of motion. Electrostatic and magnetic fields thus
correspond to states of motion in the medium, in which, how-
ever, there is no bodily flow; for the two kinds of filament
produce circulation in opposite directions.
336 Models of the Aether.
It is possible that hollow vortices are better adapted than
ordinary vortex-filaments for the construction of models of the
aether. Such, at any rate, was the opinion of Thomson (Kelvin)
in his later years.* The analytical difficulties of the subject are
formidable, and progress is consequently slow ; but among the
many mechanical schemes which have been devised to represent
electrical and optical phenomena, none possesses greater interest
than that which pictures the aether as a vortex-sponge.
* Proc. Roy. Irish Acad., November 30, 1889 ; Kelvin's Math, and Phys.
Papers, iv, p. 202. "Rotational vortex-cores," he wrote, "must he absolutely
discarded ; and we must have nothing hut irrotational revolution and vacuous
cores."
( 337 )
CHAPTEE X.
THE FOLLOWERS OF MAXWELL.
THE most notable imperfection in the electromagnetic theory
of light, as presented in Maxwell's original memoirs, was the
absence of any explanation of reflexion and refraction. Before
the publication of Maxwell's Treatise, however, a method of
supplying the omission was indicated by Helmholtz.* The
principles on which the explanation depends are that the
normal component of the electric displacement D, the tangential
components of the electric force E, and the magnetic vector B
or H, are to be continuous across the interface at which the
reflexion takes place; the optical difference between the con-
tiguous bodies being represented by a difference in their
dielectric constants, and the electric vector being assumed to
be at right angles to the plane of polarization.-)- The analysis
required is a mere transcription of MacCullagh's theory of
reflexion,| if the derivate of MacCullagh's displacement e with
respect to the time be interpreted as the magnetic force,
fi curl e as the electric force, and curl e as the electric displace-
ment. The mathematical details of the solution were not given
by Helmholtz himself, but were supplied a few years later in
the inaugural dissertation of H. A. Lorentz.§
In the years immediately following the publication of
Maxwell's Treatise, a certain amount of evidence in favour of
* Journal fur Math. Ixxii (1870), p. 68, note.
t Helmholtz (loc. cit.) pointed out that if the optical difference between the
media were assumed to be due to a difference in their magnetic permeabilities, it
would be necessary to suppose the magnetic vector at right angles to the plane of
polarization in order to obtain Fresnel's sine and tangent formulae of reflexion.
I Cf. pp. 148, 149, 154-156.
§ Zeitschrift fiir Math. u. Phys. xxii (1877), pp. 1, 205 : Over de theorie der
terugkaatsing en breking van het licht, Arnhem, 1875. Lorentz's work was based
on Helmholtz's equations, but remains substantially unchanged when Maxwell's
formulae are substituted.
Z
338 The Followers of Maxwell.
his theory was furnished by experiment. That an electric field
is closely concerned with the propagation of light was demon-
strated in 1875, when John Kerr* showed that dielectrics
subjected to powerful electrostatic force acquire the property
of double refraction, their optical behaviour being similar to
that of uniaxal crystals whose axes are directed along the lines
of force.
Other researches undertaken at this time had a more direct
bearing on the questions at issue between the hypothesis of
Maxwell and the older potential theories. In 1875-6 Helmholtzf
and his pupil Schiller^ attempted to discriminate between the
various doctrines and formulae relative to unclosed circuits by
performing a crucial experiment.
It was agreed in all theories that a ring-shaped magnet,
which returns into itself so as to have no poles, can exert no
ponderomotive force on other magnets or 011 closed electric
currents. Helmholtz§ had, however, shown in 1873 that accord-
ing to the potential-theories such a magnet would exert a
ponderomotive force on an unclosed current. The matter was
tested by suspending a magnetized steel ring by a long fibre
in a closed metallic case, near which was placed a terminal of
a Holtz machine. No ponderomotive force could be observed
when the machine was put in action so as to produce a brush
discharge from the terminal : from which it was inferred that
the potential-theories do not correctly represent the phenomena,
at least when displacement-currents and convection -currents
(such as that of the electricity carried by the electrically repelled
air from the terminal) are not taken into account.
The researches of Helmholtz and Schiller brought into
prominence the question as to the effects produced by the
* Phil. Mag. (4) 1 (1875), pp. 337, 446 ; (5) viii (1879), pp. 85, 229 ; xiii (1882),
pp. 153, 248.
t Monatsberichte d. Acad. d. Berlin/1875, p. 400. Ann. d. Phys., clviii (1876),
p. 87. t Ann. d. Phys. clix (1876), pp. 456, 537 ; clx (1877), p. 333.
\ The valuable memoirs by Helmholtz in Journal fiir Math. Ixxii (1870),
p. 57 ; Ixxv (1873), p. 35 ; Ixxviii (1874), p. 273, to which reference has already
been made, contain a full discussion of the various possibilities of the potential-
theories.
The Followers of Maxwell. 339
translatory motion of electric charges. That the convection
of electricity is equivalent to a current had been suggested
long before by Faraday.* "If," he wrote in 1838, "a baU
be electrified positively in the middle of a room and be then
moved in any direction, effects will be produced as if a current
in the same direction had existed." To decide the matter
a new experiment inspired by Helmholtz was performed by
H. A. Kowlandf in 1876. The electrified body in Kowland's
disposition was a disk of ebonite, coated with gold leaf and
capable of turning rapidly round a vertical axis between two
fixed plates of glass, each gilt on one side. The gilt faces
of the plates could be earthed, while the ebonite disk received
electricity from a point placed near its edge ; each coating of
the disk thus formed a condenser with the plate nearest to it.
An astatic needle was placed above the upper condenser-plate,
nearly over the edge of the disk; and when the disk was rotated
a magnetic field was found to be produced. This experiment,
which has since been repeated under improved conditions by
Kowland and Hutchinson,J H. Fender §, and Eichenwald,|| shows
that the " convection-current " produced by the rotation of a
charged disk, when the other ends of the lines of force are on an
earthed stationary plate parallel to it, produces the same mag-
netic field as an ordinary conduction-current flowing in a circuit
which coincides with the path of the convection-current. When
two disks forming a condenser are rotated together, the
magnetic action is the sum of the magnetic actions of each of
the disks separately. It appears, therefore, that electric charges
cling to the matter of a conductor and move with it, so far as
Rowland's phenomenon is concerned.
The first examination of the matter from the point of view
of Maxwell's theory was undertaken by J. J. Thomson,1[ in 1881.
If an electrostatically charged body is in motion, the change in
* Exper.Re*., § 1644.
t Monatsberichte d. Akad. d. Berlin, 1876, p. 211 : Ann. d. Phys. clviii (1876),
p. 487 : Annales de Chim. et de Phys. xii (1877) p. 119.
i Phil. Mag. xxvii (1889), p. 445. \ Ibid, ii (1901), p. 179 : v (1903), p. 34.
|| Ann. d. Phys. xi (1901), p. 1. H Phil. Mag. xi (1881), p. 229.
Z 2
340 The Followers of Maxwell.
the location of the charge must produce a continuous alteration
of the electric field at any point in the surrounding medium ; or,
in the language of Maxwell's theory, there must be displacement-
currents in the medium. It was to these displacement-currents
that Thomson, in his original investigation, attributed the
magnetic effects of moving charges. The particular system
which he considered was that formed by a charged spherical
conductor, moving uniformly in a straight line. It was assumed
that the distribution of electricity remains uniform over the
surface during the motion, and that the electric field in any
position of the sphere is the same as if the sphere were at
rest ; these assumptions are true so long as quantities of order
(V/c)2 are neglected, where v denotes the velocity of the sphere
and c the velocity of light.
Thomson's method was to determine the displacement-
currents in the space outside the sphere from the known
values of the electric field, and then to calculate the vector-
potential due to these displacement-currents by means of the
formula
where S' denotes the displacement-current at (x'y'zf). The
magnetic field was then determined by the equation
H = curl A.
A defect in this investigation was pointed out by Fitz Gerald,
who, in a short but most valuable note,* published a few months
afterwards, observed that the displacement-currents of Thomson
do not satisfy the circuital condition. This is most simply seen
by considering the case in which the system consists of two
parallel plates forming a condenser; if one of the plates is
fixed, and the other plate is moved towards it, the electric field
is annihilated in the space over which the moving plate travels :
this destruction of electric displacement constitutes a displace-
ment-current, which, considered alone, is evidently not a closed
* Proc. Roy. Dublin Soc., November, 1881 ; Fitz Gerald's Scientific Writings,
p. 102.
The Followers #/ Maxwell. 341
current. The defect, as Fitz Gerald showed, may be immediately
removed by assuming that a moving charge itself is to be counted
as a current-element : the total current, thus composed of the
displacement- currents and the convection-current, is circuital.
Making this correction, Fitz Gerald found that the magnetic
force due to a sphere of charge e moving with velocity v along
the axis of z is curl (0, 0, ev/r) — a formula which shows that the
displacement-currents have no resultant magnetic effect, since
the term ev/r would be obtained from the convection-current
alone.
The expressions obtained by Thomson and Fitz Gerald were
correct only to the first order of the small quantity v/c. The
effect of including terms of higher order was considered in 1889
by Oliver Heaviside,* whose solution may be derived in the
following manner : —
Suppose that a charged system is in motion with uniform
velocity v parallel to the axis of z ; the total current consists of
the displacement- cur rent E/4?rc2 where E denotes the electric
force, and the convection-current pv where p denotes the
volume-density of electricity. So the equation which connects
magnetic force with electric current may be written
E/c2 = curl H - 4:irpv.
Eliminating E between this and the equation
curl E = - H,
and remembering that H is here circuital, we have
H/c2 - V2H = 4?r curl pv.
If, therefore, a vector-potential a be defined by the equation
a/c3 - V2a = 4?rpv,
the magnetic force will be the curl of a ; and from the equation
for a it is evident that the components ax and ay are zero, and
that az is to be determined from the equation
az/c~ - V"az = 4npv.
* Phil. Mag. xxvii (1889), p. 324.
342 The Followers of Maxwell.
Now, let (x, y, £) denote coordinates relative to axes which
are parallel to the axes (a;, y, z) , and which move with the
charged bodies ; then az is a function of (x, y, £) only ; so we
have
a a , a a
5 -IT and *' "Vr
and the preceding equation is readily seen to be equivalent to
where £1 denotes (1 - v'/c2)'^. But this is simply Poisson's
equation, with & substituted for z; so the solution may be
transcribed from the known solution of Poisson's equation : it is
/L»V dx' dy d%i'
the integrations being taken over all the space in which there
are moving charges ; or
_rrr
jJJ
If the moving system consists of a single charge e at the point
5 = 0, this gives
ev
%(1 - tf sin8 0/c")* '
where sin2 0 = (a8 + y2)/r2.
It is readily seen that the lines of magnetic force due to the
moving point-charge are circles whose centres are on the line of
motion, the magnitude of the magnetic force being
ev (1 - v2/c2) sin 8
The electric force is radial, its magnitude being
r2(l - v2sin2 0/c2)f
The fact that the electric vector due to a moving point-
charge is everywhere radial led Heaviside to conclude that the
same solution is applicable when the charge is distributed over
The Followers of Maxwell. 343
a perfectly conducting sphere whose centre is at the point, the
only chaftge being that E and H would now vanish inside the
sphere. This inference was subsequently found* to be incorrect :
a distribution of electric charge on a moving sphere could in
fact not be in equilibrium if the electric force were radial, since
there would then be nothing to balance the mechanical force
exerted on the moving charge (which is equivalent to a current)
by the magnetic field. The moving system which gives rise to
the same field as a moving point-charge is not a sphere, but an
oblate spheroid whose polar axis (which is in the direction of
motion) bears to its equatorial axis the ratio (1 - tf/c*)^ : !.•)•
The energy of the field surrounding a charged sphere is
greater when the sphere is in motion than when it is at rest.
To determine the additional energy quantitatively (retaining
only the lowest significant powers of v/c), we have only to
integrate, throughout the space outside the sphere, the expression
H2/87r, which represents the electrokinetic energy per unit
volume : the result is ezv~/3a, where e denotes the charge, v the
velocity, and a the radius of the sphere.
It is evident from this result that the work required to be
done in order to communicate a given velocity to the sphere
is greater when the sphere is charged than when it is uncharged ;
that is to say, the virtual mass of the sphere is increased by an
amount 2e2/3a, owing to the presence of the charge. This may
be regarded as arising from the self-induction of the convection-
current which is formed when the charge is set in motion. It
was suggested by J. LarmorJ and by W. Wien§ that the inertia
of ordinary ponderable matter may ultimately prove to be of
this nature, the atoms being constituted of systems of electrons. ||
» By G. F. C. Searle.
t Cf. Searle, Phil. Trans, clxxxvii (1896), p. 675, and Phil. Ma?. xliv (1897),
p. 329. On the theory of the moving electrified sphere, cf. also J. J. Thomson,
Recent Researches in Elect, and Mag., p. 16; 0. Heaviside, Electrical Papers, ii,
p. 514; Electromag. Theory, i, p. 269; W. B. Morton, Phil. Mag, xli (1896),
p. 488 ; A. Schuster, Phil. Mag. xliii (1897), p. 1.
+ Phil. Trans, clxxxvi (1895), p. 697. § Arch. Neerl (3) v (1900), p. 96.
|| Experimental evidence that the inertia of electrons is purely electromagnetic
was afterwards furnished hy W. Kaufmann, Gott. Nach., 1901, p. 143 ; 1902, p 291.
344 The Followers of Maxwell.
It may, however, be remarked that this view of th«>rigin of
mass is not altogether consistent with the principle^that the
electron is an indivisible entity. For the so-called self-induction
of the spherical electron is really the mutual induction of the
convection-currents produced by the elements of electric charge
which are distributed over its surface ; and the calculation of
this quantity presupposes the divisibility of the total charge into
elements capable of acting severally in all respects as ordinary
electric charges ; a property which appears scarcely consistent
with the supposed fundamental nature of the electron.
After the first attempt of J. J. Thomson to determine the
field produced by a moving electrified sphere, the mathematical
development of Maxwell's theory proceeded rapidly. The
problems which admit of solution in terms of known functions
are naturally those in which the conducting surfaces involved
have simple geometrical forms — planes, spheres, and cylinders.*
A result which was obtained by Horace Lamb,f when
investigating electrical motions in a spherical conductor, led
to interesting consequences. Lamb found that if a spherical
conductor is placed in a rapidly alternating field, the induced
currents are almost entirely confined to a superficial layer ; and
his result was shortly afterwards generalized by Oliver Heavi-
side,| who showed that whatever be the form of a conductor
rapidly alternating currents do not penetrate far into its sub-
stance.§ The reason for this may be readily understood : it is
virtually an application of the principle|| that a perfect conductor
is impenetrable to magnetic lines of force. No perfect conductor
is known to exist ; butU if the alternations of magnetic force to
which a good conductor such as copper is exposed are very
* Cf., e.g., C. Niven, Phil. Trans, clxxii (1881), p. 307 ; H. Lamb, Phil. Trans,
clxxiv (1883), p. 519 ; J. J. Thomson, Proc. Lond. Math. Soc. xv (1884), p. 197 :
H. A. Rowland, Phil. Mag. xvii (1884), p. 413 ; J. J. Thomson, Proc. Lond. Math.
Soc. xvii (1886), p. 310; xix (1888), p. 520; and many investigations of Oliver
Heaviside, collected in his Electrical Papers.
t Loc. cit. % Electrician, Jan. 1885.
§ The mathematical theory was given hy Lord Rayleigh, Phil. Mag. xxi. (18S6),
p. 381. Cf. Maxwell's Treatise, § 689. || Cf. p. 313.
H As was first remarked by Lord Rayleigh, Phil. Mag. xiii (1882), p. 344.
The Followers of Maxwell. 345
rapid, the. conductor has not time (so to speak) to display
the impSfection of its conductivity, and the magnetic field
is therefore unable to extend far below the surface.
The same conclusion may be reached by different reasoning.*
When the alternations of the current are very rapid, the ohmic
resistance ceases to play a dominant part, and the ordinary
equations connecting electromotive force, induction, and current
are equivalent to the conditions that the currents shall be so
distributed as to make the electrokinetic or magnetic energy a
minimum. Consider now the case of a single straight wire of
circular cross-section. The magnetic energy in the space outside
the wire is the same whatever be the distribution of current in
the cross-section (so long as it is symmetrical about the centre),
since it is the same as if the current were flowing along the
central axis ; so the condition is that the magnetic energy in
the wire shall be a minimum ; and this is obviously satisfied
when the current is concentrated in the superficial layer, since
then the magnetic force is zero in the substance of the wire.
In spite of the advances which were effected by Maxwell
and his earliest followers in the theory of electric oscillations,
the gulf between the classical electrodynamics and the theory
of light was not yet completely bridged. For in all the cases
considered in the former science, energy is merely exchanged
between one body and another, remaining within the limits of a
given system ; while in optics the energy travels freely through
space, unattached to any material body. The first discovery of
a more complete connexion between the two theories was made
by Fitz Gerald, who argued that if the unification which had
been indicated by Maxwell is valid, it ought to be possible to
generate radiant energy by purely electrical means; and in
1883f he described methods by which this could be done.
Fitz Gerald's system is what has since become known as
the magnetic oscillator : it consists of a small circuit, in which
* Of. J. Stefan, Wiener, Situungsber. xcix (1890), p. 319 ; Ann. d. Phys. xli
(1890), p. 400.
t Trans. Roy. Dublin Soc. iii (1883) ; Fitz Gerald's Scient. Writings, p. 122.
346 The Followers of Maxwell.
the strength of the current is varied according to the simple
periodic law. The circuit will be supposed to be ft circle of
small area S, whose centre is the origin and whose plane is the
plane of xy ; and the surrounding medium will be supposed
to be free aether. The current may be taken to be of strength
A cos (2ni?/jF), so that the moment of the equivalent magnet
is SA cos (2irt/T). Now in the older electrodynamics, the
vector-potential due to a magnetic molecule of (vector) moment
M at the origin is (l/47r) curl (M/r), where r denotes distance
from the origin. The vector-potential due to Fitz Gerald's
magnetic oscillator would therefore be (l/47r) curl K, where K
denotes a vector parallel to the axis of z, and of magnitude
(1/r) SA cos (2-n-t/ T). The change which is involved in replacing
the assumptions of the older electrodynamics by those of
Maxwell's theory is in the present case equivalent* to retarding
the potential ; so that the vector-potential a due to the oscillator
is (l/47r) curl K where K is still directed parallel to the axis of
z, and is of magnitude
SA 27T/ r
K = - — cos — [ t —
The electric force E at any point of space is - a, and the
magnetic force H is curl a : so that these quantities may be
calculated without difficulty. The electric energy per unit
volume is E2/8?rc2 : performing the calculations, it is found that
the value of this quantity averaged over a period of the
oscillation and also averaged over the surface of a sphere of
radius r is
The part of this which is radiated is evidently that which
is proportional to the inverse square of the distance,! so the
* Cf. pp. 298, 299.
tThe other term, which is neglected, is very small compared to the term
retained, at great distances from the origin ; it is what would be obtained if the
effects of induction of the displacement-currents were neglected : i.e. it is the
energy of the forced displacement-currents which are produced directly by the
variation of the primary current, and which originate the radiating displacement-
currents.
The Followers of Maxwell. 347
average value of the radiant energy of electric type at distance
r from the oscillator is 2iTzA*S2/3c*riT* per unit volume. The
radiant energy of magnetic type ma}' be calculated in a similar
way, and is found to have the same value ; so the total radiant
energy at distance r is 47r3^42/S^/3cVT4 per unit volume;
and therefore the energy radiated in unit time is 16ir4tA'iS2/3csT*.
This is small, unless the frequency is very high ; so that
ordinary alternating currents would give no appreciable radia-
tion. Fitz Gerald, however, in the same year* indicated a
method by which the difficulty of obtaining currents of
sufficiently high frequency might be overcome: this was, to
employ the alternating currents which are produced when
a condenser is discharged.
The Fitz Gerald radiator constructed on this principle is
closely akin to the radiator afterwards developed with such
success by Hertz : the only difference is that in Fitz Gerald's
arrangement the condenser is used merely as the store of
energy (its plates being so close together that the electrostatic
field due to the charges is practically confined to the space
between them), and the actual source of radiation is the
alternating magnetic field due to the circular loop of wire:
while in Hertz's arrangement the loop of wire is abolished,
the condenser plates are at some distance apart, and the source
of radiation is the alternating electrostatic field due to their
charges.
In the study of electrical radiation, valuable help is afforded
by a general theorem on the transfer of energy in the electro-
magnetic field, which was discovered in 1884 by John Henry
Poynting.-)- We have seen that the older writers on electric
currents recognized that an electric current is associated with
the transport of energy from one place (e.g. the voltaic cell
which maintains the current) to another (e.g. an electromotor
which is worked by the current) ; but they supposed the energy
to be conveyed by the current itself within the wire, in much
* Brit. Assoc. Rep., 1883 ; FitzGerald's Scientific Writings, p. 129.
tPhil. Trans, clxxv (1884), p. 343.
348 The Followers of Maxwell.
the same way as dynamical energy is carried by water flowing
in a pipe; whereas in Maxwell's theory, the storehouse and
vehicle of energy is the dielectric medium surrounding the wire.
What Poynting achieved was to show that the flux of energy at
any place might be expressed by a simple formula in terms of
the electric and magnetic forces at the place.
Denoting as usual by E the electric force, by D the electric
displacement, by H the magnetic force, and by B the magnetic
induction, the energy stored in unit volume of the medium is*
l ED + (1/8*) BH ;
so the increase of this in unit time is (since in isotropic media
D is proportional to E, and B is proportional to H)
ED + (1/4*) HB
or E (S - i) + (1/4*) HB,
where S denotes the total current, and i the current of
conduction ; or (in virtue of the fundamental electromagnetic
equations)
- (E . i) 4. (1/4*) (E . curl H) - (1/4*) (H . curl E},
or - (E . i) - (1/4*) div [E . H].
Now (E . i) is the amount of electric energy transformed into
heat per unit volume per second; and therefore the quantity
- (1/4*) div [E . H] must represent the deposit of energy in unit
volume per second due to the streaming of energy; which
shows that the flux of energy is represented by the vector
(1/4*) [E.HJ.f This is Poynting's theorem: that the flux of
energy at any place is represented by the vector-product of the
electric and magnetic forces, divided by 4*.*
* Cf. pp. 248, 250, 282.
t Of course any circuital vector may be added. II. M. Macdonald, Electric Waves,
p. 72, propounded a form which differs from Poynting's by a non-circuital vector.
J The analogue of Poynting's theorem in the theory of the vibrations of an
isotropic elastic solid may be easily obtained ; for from the equation of motion of
an elastic solid,
p& = - (k + 4«/3) gnid div e — n curl curl e,
it follows that
• tot* + i (* + $») (div e)» + in (curl e)'} = - div W,
The Followers of Maxwell. 349
In the special case of the field which surrounds a straight
wire carrying a continuous current, the lines of magnetic force
are circles round the axis of the wire, while the lines of electric
force are directed along the wire ; hence energy must be flowing
in the medium in a direction at right angles to the axis of the
wire. A current in any conductor may therefore be regarded
as consisting essentially of a convergence of electric and magnetic
energy from the medium upon the conductor, and its trans-
formation there into other forms.
This association of a current with motions at right angles to
the wire in which it flows doubtless suggested to Poynting the
conceptions of a memoir which he published* in the following
year. When an electric current flowing in a straight wire is
gradually increased in strength from zero, the surrounding space
becomes filled with lines of magnetic force, which have the form
of circles round the axis of the wire. Poynting, adopting
Faraday's idea of the physical reality of lines of force, assumed
that these lines of force arrive at their places by moving out-
wards from the wire ; so that the magnetic field grows by a con-
tinual emission from the wire of lines of force, which enlarge
and spread out like the circular ripples from the place where a.
stone is dropped into a pond. The electromotive force which is-
associated with a changing magnetic field was now attributed
directly to the motion of the lines of force, so that wherever
electromotive force is produced by change in the magnetic field,,
or by motion of matter through the field, the electric intensity
is equal to the number of tubes of magnetic force intersected
by unit length in unit time.
A similar conception was introduced in regard to lines of
electric force. It was assumed that any change in the total
where W denotes the vector
- (k + 4w/3) div e . e + n [curl e . e] ;
and since the expression which is differentiated with respect to t represents the
sum of the kinetic and potential energies per unit volume of the solid (save for
terms which give only surface-integrals), it is seen that W is the analogue of the
Poynting vector. Cf. L. Donati, Bologna Mem. (5) vii (1899), p. 633.
* Phil. Trans, clxxvi (1885), p. 277.
350 The Followers of Maxwell.
electric induction through a curve is caused by the passage of
tubes of force in or out across the boundary ; so that whenever
magnetomotive force is produced by change in the electric field,
or by motion of matter through the field, the magnetomotive
force is proportional to the number of tubes of electric force
intersected by unit length in unit time.
Poynting, moreover, assumed that when a steady current C
flows in a straight wire, C tubes of electric force close in upon
the wire in unit time, and are there dissolved, their energy
appearing as heat. If E denote the magnitude of the electric
force, the energy of each tube per unit length is \E, so
the amount of energy brought to the wire is \CE per unit
length per unit time. This is, however, only half the energy
actually transformed into heat in the wire : so Poynting further
assumed that E tubes of magnetic force also move in per unit
length per unit time, and finally disappear by contraction to
infinitely small rings. This motion accounts for the existence
of- the electric field ; and since each tube (which is a closed ring)
contains energy of amount J(7, the disappearance of the tubes
accounts for the remaining \GE units of energy dissipated in
the wire.
The theory of moving tubes of force has been extensively
developed by Sir Joseph Thomson.* Of the two kinds of tubes
— magnetic and electric — which had been introduced by Faraday
and used by Poynting, Thomson resolved to discard the former
and employ only the latter. This was a distinct departure
from Faraday's conceptions, in which, as we have seen, great
significance was attached to the physical reality of the magnetic
lines ; but Thomson justified his choice by inferences drawn
from the phenomena of electric conduction in liquids and gases.
As will appear subsequently, these phenomena indicate that
molecular structure is closely connected with tubes of electro-
static force — perhaps much more closely than with tubes of
magnetic force ; and Thomson therefore decided to regard
* Phil. Mag. xxxi (1891), p. 149; Thomson's Recent Researches in Elect, and
Mag. (1893), chapter i.
The Followers of Maxwell. 351
magnetism as the secondary effect, and to ascribe magnetic
fields, not to the presence of magnetic tubes, but to the motion
of electric tubes. In order to account for the fact that magnetic
fields may occur without any manifestation of electric force, he
assumed that tubes exist in great numbers everywhere in space,
either in the form of closed circuits or else terminating on atoms,
and that electric force is only perceived when the tubes have a
greater tendency to lie in one direction than in another. In a
steady magnetic field the positive and negative tubes might be
conceived to be moving in opposite directions with equal
velocities.
A beam of light might, from this point of view, be regarded
simply as a group of tubes of force which are moving with the
velocity of light at right angles to their own length. Such a
conception almost amounts to a return to the corpuscular
theory ; but since the tubes have definite directions per-
pendicular to the direction of propagation, there would now
be no difficulty in explaining polarization.
The energy accompanying all electric and magnetic pheno-
mena was supposed by Thomson to be ultimately kinetic energy
of the aether ; the electric part of it being represented by rota-
tion of the aether inside and about the tubes, and the magnetic
part being the energy of the additional disturbance set up in
the aether by the movement of the tubes. The inertia of this
latter motion he regarded as the cause of induced electromotive
force.
There was, however, one phenomenon of the electromagnetic
field as yet unexplained in terms of these conceptions — namely,
the ponderomotive force which is exerted by the field on a
conductor carrying an electric current. Now any pondero-
motive force consists in a transfer of mechanical momentum
from the agent which exerts the force to the body which
experiences it ; and it occurred to Thomson that the pondero-
motive forces of the electromagnetic field might be explained if
the moving tubes of force, which enter a conductor carrying a
current and are there dissolved, were supposed to possess
352 The Followers of Maxwell.
mechanical momentum, which could be yielded up to the
conductor. It is readily seen that such momentum must be
directed at right angles to the tube and to the magnetic
induction — a result which suggests that the momentum stored
in unit volume of the aether may be proportional to the vector-
product of the electric and magnetic vectors.
For this conjecture reasons of a more definite kind may be
given.* We have already seenf that the ponderomotive forces
on material bodies in the electromagnetic field may be accounted
for by Maxwell's supposition that across any plane in the aether
whose unit normal is N, there is a stress represented by
PN = (D . N) E - J (D .E)N + (l/47r) (B . H)H - (I/Sir) (B. H) N.
So long as the field is steady (i.e. electrostatic or magnetostatic)
the resultant of the stresses acting on any element of volume of
the aether is zero, so that the element is in equilibrium. But
when the field is variable, this is no longer the case. The
resultant stress on the aether contained within a surface S is
JJ PN . dS
integrated over the surface : transforming this into a volume-
integral, the term (D . N) E gives a term div D . E + (D . V) E,
where V denotes the vector operator (9/9a?, d/dy, d/dz) ; and the
first of these terms vanishes, since D is a circuital vector;
the term - J (D . E) N gives in the volume-integral a term
J grad (D . E) ; and the magnetic terms give similar results.
So the resultant force on unit- volume of the aether is
(D . V) E + J grad (D . E) + (l/4ir) (B . V) E + (I /Sir) grad (B . H),
which may be written
[curl E . D] + (l/47r) [curl H . B] ;
* The hypothesis that the aether is a storehouse of mechanical momentum,
which was first advanced by ,T. J. Thomson (Recent Researches in Elect, and Mag.
(1893), p. 13), was afterwards developed by H. Poincare, Archives Neerl. (2) v
(1900), p. 252, and by M. Abraham, Gott, Nach., 1902, p. 20.
tCf. p. 302.
The Followers of Maxwell. 353
or, by virtue of the fundamental equations for dielectrics,
[- B . D] + [D . B] , or (a/ft) [D . B].
This result compels us to adopt one of three alternatives:
either to modify the theory so as to reduce to zero the resultant
force on an element of free aether ; this expedient has not met
with general favour ;* or to assume that the force in question
sets the aether in motion: this alternative was chosen by
Helmholtz,f but is inconsistent with the theory of the aether
which was generally received in the closing years of the century;
or lastly, with Thomson^ to accept the principle that the aether
is itself the vehicle of mechanical momentum, of amount [D . B]
per unit volume.
Maxwell's theory was now being developed in ways which
could scarcely have been anticipated by its author. But although
every year added something to the superstructure, the founda-
tions remained much as Maxwell had laid them ; the doubtful
argument by which he had sought to justify the introduction
of displacement- currents was still all that was offered in their
defence. In 1884, however, the theory was established§ on a
different basis by a pupil of Helmholtz', Heinrich Hertz
(b. 1857, d. 1894).
The train of Hertz' ideas resembles that by which Ampere,
on hearing of Oersted's discovery of the magnetic field produced
by electric currents, inferred that electric currents should exert
ponderomotive forces on each other. Ampere argued that a
current, being competent to originate a magnetic field, must be
equivalent to a magnet in other respects ; and therefore that
currents, like magnets, should exhibit forces of mutual attraction
and repulsion.
* It was, however, adopted by G. T. "Walker, Aberration and the Electromagnetic
Field, Camb., 1900.
t Berlin Sitzungsberichte, 1893, p. 649; Ann. d. Phys. liii (1894), p. 135.
Helmholtz supposed the aether to behave as a frictionless incompressible fluid.
+ Loc. cit.
§ Ann. d. Phys. xxiii (1884), p. 84: English version in Hertz's Miscellaneous
Papers, translated by D. E. Jones and G. A. Schott, p. 273.
2 A
354 The Followers of Maxwell.
Ampere's reasoning rests on the assumption that the mag-
netic field produced by a current is in all respects of the same
nature as that produced by a magnet ; in other words, that only
one land of magnetic force exists. This principle of the " unity
of magnetic force" Hertz now proposed to supplement by assert-
ing that the electric force generated by a changing magnetic
field is identical in nature with the electric force due to electro-
static charges; this second principle he called the "unity of
electric force." Suppose, then, that a system of electric currents
i exists in otherwise empty space. According to the older
theory, these currents give rise to a vector-potential a, , equal
to Pot i ;* and the magnetic force Ht is the curl of at : while
the electric force E! at any point in the field, produced by the
variation of the currents, is — ai.
It is now assumed that the electric force so produced is
indistinguishable from the electric force which would be set
up by electrostatic charges, and therefore that the system of
varying currents exerts ponderomobive forces on electrostatic
charges ; the principle of action and reaction then requires that
electrostatic charges should exert ponderomotive forces on a
system of varying currents, and consequently (again appealing
to the principle of the unity of electric force) that two systems
of varying currents should exert on each other ponderomotive
forces due to the variations.
But just as Helmholtz,f by aid of the principle of conser-
vation of energy, deduced the existence of an electromotive
force of induction from the existence of the ponderomotive
forces between electric currents (Le. variable electric systems),
so from the existence of ponderomotive forces between variable
systems of currents (i.e. variable magnetic systems) we may
infer that variations in the rate of change of a variable magnetic
system give rise to induced magnetic forces in the surrounding
space. The analytical formulae which determine these forces
* a = Pot /3 is used to denote the solution of the equation V'a + 47r£ = 0.
fCf. p. 243.
The Followers of Maxwell. 355
will be of the same kind as in the electric case ; so that the
induced magnetic force H' is given by an equation of the form
where c denotes some constant, and bi, which is analogous to
the vector-potential in the electric case, is a circuital vector
whose curl is the electric force E! of the variable magnetic
system. The value of bi is therefore (l/47r) curl Pot Et : so
we have
H' = - J-. |, curl Pot a,
47TC" (jt~
This must be added to Hi. Writing H2 for the sum, Hi + H', we
see that H2 is the curl of a2, where
and the electric force E2 will then be - a2.
This system is not, however, final ; for we must now perform
the process again with these improved values of the electric
and magnetic forces and the vector-potential ; and so we obtain
for the magnetic force the value curl a3, and for the electric
force the value - a3, where
1 r)z 1 ^*
= ax - - Pot ax + — — -- — Pot Pot
4rrc2 fit*
This process must again be repeated indefinitely ; so finally we
obtain for the magnetic force H the value curl a, and for the
electric force E the value - a, where
1 £}*>
- Pot Pot Pot a! +
(47TC2)3
2A2
356 The Followers of Maxwell.
It is evident that the quantity a thus defined satisfies the
equation
or v*a - -— a = - 47ri.
c2 dt'
This equation may be written
while the equations H = curl a, E = - a give
curl E = - H.
These are, however, the fundamental equations of Maxwell's
theory in the form given in his memoir of 1868,*
That Hertz's deduction is ingenious and interesting will
readily be admitted. That it is conclusive may scarcely be
claimed : for the argument of Helmholtz regarding the induc-
tion of currents is not altogether satisfactory; and Hertz, in
following his master, is on no surer ground.
In the course of a discussion^ on the validity of Hertz's
assumptions, which followed the publication of his paper,
E. AulingerJ brought to light a contradiction between the
principles of the unity of electric and of magnetic force and
the electrodynamics of Weber. Consider an electrostatically
charged hollow sphere, in the interior of which is a wire
carrying a variable current. According to Weber's theory,
the sphere would exert a turning couple on the wire; but
according to Hertz's principles, no action would be exerted,
since charging the sphere makes no difference to either the
electric or the magnetic force in its interior. The experiment
thus suggested would be a crucial test of the correctness of
Weber's theory ; it has the advantage of requiring nothing
but closed currents and electrostatic charges at rest ; but
the quantities to be observed would be on the limits of
observational accuracy.
»Cf. p. 287.
f Lorberg, Ann. d. Phys. xxvii (1886), p. 666; xxxi (1887), p. 131.
Boltzmann, ibid, xxix (1886), p. 598. + Ann. d. Phys. xxvii (1886), p. 119.
The Followers of Maxwell. 357
After his attempt to justify the Maxwellian equations on
theoretical grounds, Hertz turned his attention to the possibility
of verifying them by direct experiment. His interest in the
matter had first been aroused some years previously, when the
Berlin Academy proposed as a prize subject " To establish
experimentally a relation between electromagnetic actions and
the polarization of dielectrics." Helmholtz suggested to Hertz
that he should attempt the solution ; but at the time he saw
no way of bringing phenomena of this kind within the limits of
observation. From this time forward, however, the idea of electric
oscillations was continually present to his mind ; and in the
spring of 1886 he noticed an effect* which formed the starting-
point of his later researches. When an open circuit was formed
of a piece of copper wire, bent into the form of a rectangle,
so that the ends of the wire were separated only by a short air-
gap, and when this open circuit was connected by a wire with
any point of a circuit through which the spark -discharge of an
induction-coil was taking place, it was found that a spark
passed in the air-gap of the open circuit. This was explained
by supposing that the change of potential, which is propagated
along the connecting wire from the induction-coil, reaches one
end of the open circuit before it reaches the other, so that a
spark passes between them; and the phenomenon therefore
was regarded as indicating a finite velocity of propagation of
electric potential along wires.!
* Ann. d. Phys. xxxi (1887), p. 421. Hertz's Electric Waves, translated by
D. E. Jones, p. 29.
t Unknown to Hertz, the transmission of electric waves along wires had been
observed in 1870 by Wilhelm von Bezold, Miinchen Sitzungsbericlite, i (1870),
p. 113 ; Phil. Mag. xl (1870), p. 42. «* If," he wrote at the conclusion of a series
of experiments, "electrical waves be sent into a wire insulated at the end, they
will be reflected at that end. The phenomena which accompany this process in
alternating discharges appear to owe their origin to the interference of the
advancing and reflected waves," and, "an electric discharge travels with the
«atne rapidity in wires of equal length, without reference to the materials of
which these wires are made."
The subject was investigated by 0. J. Lodge and A. P. Chattock at almost the
same time as Hertz's experiments were being carried out: mention was made of
their researches at the meeting of the British Association in 1888.
358 The Followers of Maxwell.
Continuing his experiments, Hertz* found that a spark
could be induced in the open or secondary circuit even when it
was not in metallic connexion with the primary circuit in which
the electric oscillations were generated; and he rightly inter-
preted the phenomenon by showing that the secondary circuit
was of such dimensions as to make the free period of electric
oscillations in it nearly equal to the period of the oscillations
in the primary circuit ; the disturbance which passed from one
circuit to the other by induction would consequently be greatly
intensified in the secondary circuit by resonance.
The discovery that sparks may be produced in the air-gap
of a secondary circuit, provided it has the dimensions proper
for resonance, was of great importance : for it supplied a method
of detecting electrical effects in air at a distance from the primary
disturbance ; a suitable detector was in fact all that was needed
in order to observe the propagation of electric waves in free
space, and thereby decisively test the Maxwellian theory. To
this work Hertz now addressed himself.f
The radiator or primary source of the disturbances studied
by Hertz may be constructed of two sheets of metal in the
same plane, each sheet carrying a stiff wire which projects
towards the other sheet and terminates in a knob ; the sheets
are to be excited by connecting them to the terminals of an
induction coil. The sheets may be regarded as the two coatings
of a modified Leyden jar, with air as the dielectric between
them ; the electric field is extended throughout the air, instead
of being confined to the narrow space between the coatings, as
in the ordinary Leyden jar. Such a disposition ensures that
the system shall lose a large part of its energy by radiation
at each oscillation.
* Loc. cit.
t Sir Oliver Lodge was about this time independently studying electric oscilla-
tions in air in connexion with the theory of lightning-conductors : cf. Lodge,
Phil. Mag. xxvi (1888), p. 217. So long before as 1842, Joseph Henry, of
Washington, had noticed that the inductive effects of the Leyden jar discharge
could be observed at considerable distances, and had even suggested a comparison
with " a spark from flint and steel in the case of light."
The Followers of Maxwell. 359
As in the jar discharge,* the electricity surges from one
sheet to the other, with a period proportional to (CL)l, where
0 denotes the electrostatic capacity of the system formed by
the two sheets, and L denotes the self-induction of the
connexion. The capacity and induction should be made as
small as possible in order to make the period small. The
detector used by Hertz was that already described, namely,
a wire bent into an incompletely closed curve, and of such
dimensions that its free period of oscillation was the same
as that of the primary oscillation, so that resonance might take
place.
Towards the end of the year 1887, when studying the sparks
induced in the resonating circuit by the primary disturbance,
Hertz noticedf that the phenomena were distinctly modified
when a large mass of an insulating substance was brought
into the neighbourhood of the apparatus ; thus confirming the
principle that the changing electric polarization which is pro-
duced when an alternating electric force acts on a dielectric
is capable of displaying electromagnetic effects.
Early in the following year (1888) Hertz determined to
verify Maxwell's theory directly by showing that electro-
magnetic actions are propagated in air with a finite velocity .{
For this purpose he transmitted the disturbance from the
primary oscillator by two different paths, viz., through the air
and along a wire ; and having exposed the detector to the joint
influence of the two partial disturbances, he observed inter-
ference between them. In this way he found the ratio of the
velocity of electric waves in air to their velocity when conducted
by wires ; and the latter velocity he determined by observing
the distance between the nodes of stationary waves in the wire,
and calculating the period of the primary oscillation. The
velocity of propagation of electric disturbances in air was in
* Cf. p. 253.
t Ann. d. Phys. xxxiv, p. 373. Electric Waves (English edition), p. 95.
J Ann. d. Phys. xxxiv (1888), p. 551. Electric Waves (English edition) p. 107.
360 The Followers of Maxwell.
this way shown to be finite and of the same order as the
velocity of light.*
Later in 1888 Hertzf showed that electric waves in air are
reflected at the surface of a wall ; stationary waves may thus
be produced, and interference may be obtained between direct
and reflected beams travelling in the same direction.
The theoretical analysis of the disturbance emitted by a
Hertzian radiator according to Maxwell's theory was given by
Hertz in the following year.J
The effects of the radiator are chiefly determined by the
free electric charges which, alternately appearing at the two
sides, generate an electric field by their presence and a magnetic
field by their motion. In each oscillation, as the charges on
the poles of the radiator increase from zero, lines of electric
force, having their ends on these poles, move outwards into
the surrounding space. When the charges on the poles attain
their greatest values, the lines cease to issue outwards, and the
existing lines begin to retreat inwards towards the poles; but
the outer lines of force contract in such a way that their upper
and lower parts touch each other at some distance from the
radiator, and the remoter portion of each of these lines thus
takes the form of a loop ; and when the rest of the line of
force retreats inwards towards the radiator, this loop becomes
detached and is propagated outwards as radiation. In this
way the radiator emits a series of whirl-rings, which as they
move grow thinner and wider; at a distance, the disturbance
* Hertz's experiments gave the value 45/28 for the ratio of the velocity of
electric waves in air to the velocity of electric waves conducted by the wires, and
2 x 1010 cms. per sec. for the latter velocity. These numbers were afterwards
found to be open to objection: Poincare (Comptes Rendus, cxi (1890), p. 322)
showed that the period calculated by Hertz was V2 x the true period, which would
make the velocity of propagation in air equal to that of light x v'2. Ernst Lecher
(Wiener Berichte, May 8, 1890; Phil. Mag. xxx (1890), p. 128), experimenting
on the velocity of propagation of electric vibrations in wires, found instead of
Hertz's 2 x 1010 cms. per sec., a value within two per cent, of the velocity of
light. E. Sarasin and L. De La Rive at Geneva (Archives des Sc. Phys. xxix (1893))
finally proved that the velocities of propagation in air and along wires are equal.
t Ann. d. Phys. xxxiv (1888), p. 610. Electric Waves (English edition), p. 124.
J Ibid., xxxvi (1889), p. 1. Electric Waves (English edition), p. 137.
The Followers of Maxwell. 361
is approximately a plane wave, the opposite sides of the ring
representing the two phases of the wave. When one of these
rings has become detached from the radiator, the energy con-
tained may subsequently be regarded as travelling outwards
with it.
To discuss the problem analytically* we take the axis of
the radiator as axis of z, and the centre of the spark-gap as
origin. The field may be regarded as due to an electric doublet
formed of a positive and an equal negative charge, displaced
from each other along the axis of the vibrator, and of
moment
Ae~p^ sin (2irct/\),
the factor e~p^ being inserted to represent the damping.
The simplest method of proceeding, which was suggested by
Fitz Gerald,f is to form the retarded potentials <£ and a of
L. Lorenz.J These are determined in terms of the charges and
their velocities by the equations
I. * = a<^, o.-s^,
whence it is readily shown that in the present case
4> = - dF/dz, a = (0, 0,
where
Ae'KC-'P , 2;r . ,
F = sin — (ct - r).
T A*
The electric and magnetic forces are then determined by the
equations
E = c2 grad <£ - a, H = curl a.
It is found that the electric force may be regarded as com-
pounded of a force <£2, parallel to the axis of the vibrator and
depending at any instant only on the distance from the vibrator,
together with a force fa sin 0 acting in the meridian plane
* Cf. Karl Pearson and A. Lee, Phil. Trans, cxciii (1899), p. 165.
+ Brit. Assoc. Rep., Leeds (1890), p. 755.
J Cf. p. 298. The use of retarded potentials was also recommended in the
following year by Poincare, Comptes Rendus, cxiii (1891), p. 515.
362 The Followers of Maxwell.
perpendicular to the radius from the centre, where $1 depends at
any instant only on the distance from the vibrator, and 0
denotes the angle which the radius makes with the axis of the
oscillator. At points on the axis, and in the equatorial plane,
the electric force is parallel to the axis. At a great distance
from the oscillator, 02 is small compared with 0,, so the wave is
purely transverse. The magnetic force is directed along circles
whose centres are on the axis of the radiator ; and its magnitude
may be represented in the form 03 sin 9, where 03 depends
only on r and t ; at great distances from the radiator, c<£3 is
approximately equal to 0,.
If the activity of the oscillator be supposed to be continually
maintained, so that there is no damping, we may replace p{ by
zero, and may proceed as in the case of the magnetic oscillator*
to determine the amount of energy radiated. The mean out-
ward flow of energy per unit time is found to be Jc3^2 (27T/X)4;
from which it is seen that the rate of loss of energy by radiation
increases greatly as the wave-length decreases.
The action of an electrical vibrator may be studied by the
aid of mechanical models. In one of these, devised by Larmor,f
the aether is represented by an incompressible elastic solid, in
which are two cavities, corresponding to the conductors of the
vibrator, filled with incompressible fluid of negligible inertia.
The electric force is represented by the displacement of the
solid. For such rapid alternations as are here considered,
the metallic poles behave as perfect conductors; and the
tangential components of electric force at their surfaces' are
zero. This condition may be satisfied in the model by suppos-
ing the lining of each cavity to be of flexible sheet-metal, so as
to be incapable of tangential displacement ; the normal displace-
ment of the lining then corresponds to the surface-density of
electric charge on the conductor.
In order to obtain oscillations in the solid resembling those
of an electric vibrator, we may suppose that the two cavities
* Cf. p. 346.
7 Proc. Camb. Phil. Soc. vii (1891), p. 165.
The Followers of Maxwell. 36li
have the form of semicircular tubes forming the two halves
of a complete circle. Each tube is enlarged at each of
its ends, so as to present a front of considerable area to the
corresponding front at the end of the other tube. Thus at each
end of one diameter of the circle there is a pair of opposing
fronts, which are separated from each other by a thin sheet
of the elastic solid.
The disturbance may be originated by forcing an excess of
liquid into one of the enlarged ends of one of the cavities. This
involves displacing the thin sheet of elastic solid, which
separates it from the opposing front of the other cavity, and
thus causing a corresponding deficiency of liquid in the enlarged
end behind this front. The liquid will then surge backwards
and forwards in each cavity between its enlarged ends ; and,
the motion being communicated to the elastic solid, vibrations
will be generated resembling those which are produced in the
aether by a Hertzian oscillator.
In the latter part of the year 1888 the researches of Hertz*
yielded more complete evidence of the similarity of electric
waves to light. It was shown that the part of the radiation
from an oscillator which was transmitted through an opening in
a screen was propagated in a straight line, with diffraction effects.
Of the other properties of light, polarization existed in the
original radiation, as was evident from the manner in which it
was produced ; and polarization in other directions was obtained
by passing the waves through a grating of parallel metallic wires ;
the component of the electric force parallel to the wires was
absorbed, so that in the transmitted beam the electric vibration
was at right angles to the wires. This effect obviously resembled
the polarization of ordinary light by a plate of tourmaline.
Refraction was obtained by passing the radiation through
prisms of hard pitch. j-
* Ann. d. Phys.xxxvi (1889), p. 769; Electric Waves (Englished.), p. 172.
I 0. J. Lodge and J. L. Ho ward in the same year showed that electric radiation
might be refracted and concentrated hy means of large lenses. Cf. Phil. Mag.
xxvii (1889), p. 48.
364 The Followers of Maxwell.
The old question as to whether the light-vector is in, or at
right angles to, the plane of polarization* now presented itself
in a new aspect. The wave-front of an electric wave contains
two vectors, the electric and magnetic, which are at right angles
to each other. Which of these is in the plane of polarization ?
The answer was furnished by Fitz Gerald and Trouton,f who
found on reflecting Hertzian waves from a wall of masonry that
no reflexion was obtained at the polarizing angle when the
vibrator was in the plane of reflexion. The inference from this
is that the magnetic vector is in the plane of polarization of the
electric wave, and the electric vector is at right angles to the
plane of polarization. An interesting development followed in
1890, when 0. Wiener^ succeeded in photographing stationary
waves of light. The stationary waves were obtained by the
composition of a beam incident on a mirror with the reflected
beam, and were photographed on a thin film of transparent
collodion, placed close to the mirror and slightly inclined to it.
If the beam used in such an experiment is plane-polarized, and
is incident at an angle of 45°, the stationary vector is evidently
that perpendicular to the plane of incidence; but Wiener
found that under these conditions the effect was obtained only
when the light was polarized in the plane of incidence ; so
that the chemical activity must be associated with the vector
perpendicular to the plane of polarization — i.e., the electric
vector.
In 1890 and the years immediately following appeared
several memoirs relating to the fundamental equations of
electro-magnetic theory. Hertz, after presenting§ the general
* Cf. pp. 168 et sqq.
f Nature, xxxix (1889), p. 391.
\ Ann. d. Phys. xl (1890), p. 203. Cf. a controversy regarding the results ;
Comptes Rendus, cxii (1891), pp. 186, 325, 329, 365, 383, 456 ; and Ann. d. Phys.
xli (1890), p. 154 ; xliii(1891), p. 177; xlviii (1893), p. 119.
§ Gott. Nach. 1890, p. 106; Aim. d. Phys. xl (1890), p. 577; Electric Waves
(English ed.), p. 195. In this memoir Hertz advocated the form of the equations
which Maxwell had used in his paper of 1868 (cf. supra, p. 287) in preference to
the earlier form, which involved the scalar and vector potentials.
The Followers of Maxwell. 365
content of Maxwell's theory for bodies at rest, proceeded* to
extend the equations to the case in which material bodies are
in motion in the field.
In a really comprehensive and correct theory, as Hertz
remarked, a distinction should be drawn between the quantities
which specify the state of the aether at every point, and those
which specify the state of the ponderable matter entangled with
it. This anticipation has been fulfilled by later investigators ;
but Hertz considered that the time was not ripe for such a
complete theory, and preferred, like Maxwell, to assume that
the state of the compound system — matter plus aether — can be
specified in the same way when the matter moves as when it is
at rest ; or, as Hertz himself expressed it, that " the aether
contained within ponderable bodies moves with them."
Maxwell's own hypothesis with regard to moving systemsf
amounted merely to a modification in the equation
B = - curl E,
which represents the law that the electromotive force in a
closed circuit is measured by the rate of decrease in the number
of lines of magnetic induction which pass through the circuit.
This law is true whether the circuit is at rest or in motion ; but
in the latter case, the E in the equation must be taken to be the
electromotive force in a stationary circuit whose position
momentarily coincides with that of the moving circuit; and
since an electromotive force [w . B] is generated in matter by
its motion with velocity w in a magnetic field B, we see that E
is connected with the electromotive force E' in the moving
ponderable body by the equation
E' = E + [w . B],
so that the equation of electromagnetic induction in the moving
body is
B = - curl E' + curl [w . B].
* Ann. d. Phys. xli (1890), p. 369 ; Electric Waves (English ed.), p. 241.
The propagation of light through a moving dielectric had been discussed
previously, on the basis of Maxwell's equations for moving bodies, by J. J. Thomson,
Phil. Mag. ix (1880), p. 284 ; Proc. Camb. Phil. Soc. v (1885), p. 250.
tCf. p. 288.
366 The Followers of Maxwell.
Maxwell made no change in the other electromagnetic
equations, which therefore retained the customary forms
D = f E'/47rc2, div D = 0, 47r(i 4 D) = curl H,
Hertz, however, impressed by the duality of electric and
magnetic phenomena, modified the last of these equations by
assuming that a magnetic force 4?r [D . w] is generated in a
dielectric which moves with velocity w in an electric field ; such
a force would be the magnetic analogue of the electromotive
force of induction. A term involving curl |D . w] is then
introduced into the last equation.
The theory of Hertz resembles in many respects that of
Heaviside,* who likewise insisted much on the duplex nature
of the electromagnetic field, and was in consequence disposed
to accept the term involving curl [D . w] in the equations of
moving media. Heaviside recognized more clearly than his
predecessors the distinction between the force E', which
determines the flux D, and the force E, whose curl represents
the electric current ; and, in conformity with his principle of
duality, he made a similar distinction between the magnetic
force H', which determines the flux B, and the force H, whose
curl represents the " magnetic current." This distinction, as
Heaviside showed, is of importance when the system is
acted on by " impressed forces," such as voltaic electromotive
forces, or permanent magnetization; these latter must be
included in E' and H', since they help to give rise to the fluxes
D and B ; but they must not be included in E and H, since their
curls are not electric or magnetic currents ; so that in general
we have
E' = E + e, H' = + h,
where e and h denote the impressed forces.
Developing the theory by the aid of these conceptions,
Heaviside was led to make a further modification. An im-
* Heaviside's general theory was published in a series of papers in the
Electrician, from 1885 onwards. His earlier work was republished in his
Electrical Papers (2 vols., 1892), and his Electromagnetic Theory (2 vols., 1894).
Mention may be specially made of a memoir in Phil. Trans, clxxxiii (1892),
p. 423.
The Followers of Maxwell. 367
pressed force is best defined in terms of the energy which it
communicates to the system ; thus, if e be an impressed electric
force, the energy communicated to unit volume of the electro-
magnetic system in unit time is e x the electric current.
In order that this equation may be true, it is necessary to
regard the electric current in a moving medium as composed
of the conduction-current, displacement-current, convection-
current, and also of the term curl [D . w] , whose presence in
the equation we have already noticed. This may be called
the current of dielectric convection. Thus the total current is
S = D + i + pw + curl [D . w] ,
where pw denotes the conduction-current ; and the equation
connecting current with magnetic force is
curl (H' - h0) = 4?rS,
where h0 denotes the impressed magnetic forces other than that
induced by motion of the medium.
We must now consider the advances which were effected
during the period following the publication of Maxwell's
Treatise in some of the special problems of electricity and
optics.
We have seen* that Maxwell accounted for the rotation of
the plane of polarization of light in a medium subjected to a
magnetic field K by adding to the kinetic energy of the aether,
which is represented by Jpe*, a term J<r (e . curl 9e/90), where
cr is a magneto-optic constant characteristic of the substance
through which the light is transmitted, and d/dO stands for
Kxdl'dx + Kydldy + K-d/dz. This theory was developed further
in 1879 by Fitz Gerald,f who brought it into closer connexion
with the electromagnetic theory of light by identifying the curl
of the displacement e of the aethereal particles with the electric
displacement ; the derivate of e with respect to the time then
corresponds to the magnetic force. Being thus in possession of
a definitely electromagnetic theory of the magnetic rotation of
* Cf . p. 308.
t Phil. Trans., 1879, p. 691. Fitz Gerald's Scient. Writings, p. 45.
368 The Followers of Maxwell.
light, Fitz Gerald proceeded to extend it so as to take account
of a closely related phenomenon. In 1876 J. Kerr* had shown
experimentally that when plane-polarized light is regularly
reflected from either pole of an iron electromagnet, the reflected
ray has a component polarized in a plane at right angles to the
ordinary reflected ray. Shortly after this discovery had been
made known, Fitz Geraldf had proposed to explain it by means
of the same term in the equations which accounts for the mag-
netic rotation of light in transparent bodies. His argument was
that if the incident plane-polarized ray be resolved into two
rays circularly polarized in opposite senses, the refractive index
will have different values for these two rays, and hence the
intensities after reflexion will be different; so that on re-
compounding them, two plane-polarized rays will be obtained —
one polarized in the plane of incidence, and the other polarized
at right angles to it.
The analytical discussion of Kerr's phenomenon, which was
given by Fitz Gerald in his memoir of 1879, was based on these
ideas ; the most essential features of the phenomenon were
explained, but the investigation was in some respects imperfect.}
Anew and fruitful conception was introduced in 1879-1880,
when H. A. Eowland§ suggested a connexion between the
magnetic rotation of light and the phenomenon which had been
discovered by his pupil Hall.|| Hall's effect may be regarded
* Phil. Mag. (5) iii (1877), p. 321.
t Proc. 11. S. xxv (1877), p. 447 ; Fitz Gerald's Sclent. Writings, p. 9.
J Cf. Larmor's remarks in his Report on the Action oj Magnetism on Light,
Brit. Assoc. Kep., 1893 ; and his editorial comments in Fitz Gerald's Scientific
Writings. Larmor traced to its source an inconsistency in the equations hy which
Fitz Gerald had represented the boundary-conditions at an interface between the
media. Fitz Gerald had indeed made the mistake, similar to that which was so often
made hy the earlier writers on the elastic-solid theory of light, of forgetting that when
a medium is assumed to be incompressible, the condition of in compressibility must
be introduced into the variational equation of motion (as was done supra, p. 172).
Larmor showed that when this correction was made, new terms (resembling the
terms in p, supra, p. 172) made their appearance; and the inconsistency in the
equations was thus removed.
§ Amer. Jour. Math, ii, p. 354, iii, p. 89; Phil. Mag. xi (1881), p. 254.
|| Cf. p. 321.
The Followers of Maxwell. 369
as a rotation of conduction-currents under the influence of a
magnetic field ; and if it be assumed that displacement-currents
in dielectrics are rotated in the same way, the Faraday effect
may evidently be explained. Considering the matter from the
analytical point of view, the Hall effect may be represented by
the addition of a term k [K . S] to the electromotive force,
where K denotes the impressed magnetic force, and S denotes
the current : so Kowland assumed that in dielectrics there is an
additional term in the electric force, proportional to [K . D], i.e.
proportional to the rate of increase of [K . D]. Now it is
universally true that the total electric force round a circuit is
proportional to the rate of decrease of the total magnetic
induction through the circuit : so the total magnetic induction
through the circuit must contain a term proportional to the
integral of [K . D] taken round the circuit : and therefore the
magnetic induction at any point must contain a term proportional
to curl [K . D]. We may therefore write
B = H + a curl [K . D],
where <r denotes a constant. But if this be combined with the
customary electromagnetic equations
curl H = 47rD, curl E = - B, D = eE/47rc3,
and all the vectors except B be eliminated (K being treated
as a constant), we obtain the equation
B - (c7/0 V2B + O/47r) curl
where 3/80 stands for (Kxd/fa + Kyd/dy + Kzd/dz) ; and this is
identical with the equation which Maxwell had given* for the
motion of the aether in magnetized media. It follows that the
assumptions of Maxwell and of Eowland, different though they
are physically, lead to the same analytical equations— at any
rate so far as concerns propagation through a homogeneous
medium.
The connexions of Hall's phenomenon with the magnetic
rotation of light, and with the reflexion of light from magnetized
* Cf. p. 308.
2 B
370 The Followers of Maxwell.
metals, were extensively studied* in the years following the
publication of Kowland's memoir: but it was not until the
modern theory of electrons had been developed that a satisfactory
representation of the molecular processes involved in magneto-
optic phenomena was attained.
The allied phenomenon of rotary polarization in naturally
active bodies was investigated in 1892 by Goldhammer.f It
* The theory of Basset (Phil. Trans, clxxxii (1891), p. 371) was, like Rowland's,
based on the idea of extending Hall's phenomenon to dielectric media. An objec-
tion to this theory was that the tangential component of the electromotive force
was not continuous across the interface between a magnetized and an unmagnetized
medium ; but Basset subsequently overcame this difficulty (Nature, Hi (1 895), p. 618 ;
liii (1895), p. 130; Amer. Jour. Math, xix (1897), p. 60)— the effect analogous to
Hall's being introduced into the equation connecting electric displacement with
electric force, so that the equation took the form
E = (47rc2/€) D + ff [K . D].
Basset, in 1893 (Proc. Camb. Phil. Soc. viii, p. 68), derived analytical
expressions which represent Kerr's magneto-optic phenomenon by substituting u
complex quantity for the refractive index in the formulae applicable to transparent
magnetized substances.
The magnetic rotation of light and Kerr's phenomenon have been investigated
also by R. T. Glazebrook, Phil. Mag. xi (1881), p. 397 ; by J. J. Thomson,
Recent Researches, p. 482 : by D. A. Goldhammer, Ann. d. Phys. xlvi (1892),
p. 71 ; xlvii (1892), p. 345; xlviii (1893), p. 740; 1 (1893), p. 772 : by P. Drude,
Ann. d. Phys. xlvi (1892), p. 353; xlviii (1893), p. 122; xlix (1893), p. 690;
lii (1894)) p. 496 : by C. H. Wind, Verslagen Kon. Akad. Amsterdam, 29th Sept.,
1894 : by Reiff, Ann. d. Phys. Ivii (1896), p. 281 : by J. G. Leathern, Phil.
Trans, cxc (1897), p. 89; Trans. Camb. Phil. Soc. xvii (1898), p. 16: and by
W. Voigt in many memoirs, and in his treatise, Magneto- und Elektro-optik.
Larmor's report presented to the British Association in 1893 has been already
mentioned.
In most of the later theories the equations of propagation of light in magnetized
metals are derived from the two fundamental electromagnetic equations
curl H = 4?rS, - curl E = H ;
the total current S being assumed to consist of a part (the displacement-current)
proportional to E, a part (the conduction -current) proportional to E, and a part
proportional to the vector-product of E and the magnetization.
Various mechanical models of media in which magneto-optic phenomena take
place have been devised at different times. W. Thomson (Proc. Lond. Math. Soc.
vi (1875)) investigated the propagation of waves of displacement along a stretched
chain whose links contain rotating fly-wheels : cf . also Larmor, Proc. Lond. Math.
Soc. xxi. (1890), p. 423 ; xxiii (1891), p. 127 ; F. Hasenohrl, Wien Sitzungsberichte
cvii, 2a (189S), p. 1015 ; W. Thomson (Kelvin), Phil. Mag. xlviii (1899), p. 236,
and Baltimore Lectures ; and Fitz Gerald, Electrician, Aug. 4, 1899, Fitz Gerald's
Scientific Writings, p. 481. t Journal de Physique (3) i, pp. 205, 345.
The Followers of Maxwell. 371
will be remembered* that in the elastic-solid theory of
Boussinesq, the rotation of the plane of polarization of
saccharine solutions had been represented by substituting the
equation
e' = Ae + B curl e
in place of the usual equation
e' = Ae.
Goldhammer now proposed to represent rotatory power in the
electromagnetic theory by substituting the equation
E = (4ircVO D + k curl D,
in place of the customary equation
E = (4ircVO D :
the constant k being a measure of the natural rotatory power
of the substance concerned. The remaining equations are as
usual.
curl H = 47rD, - curl E = H
Eliminating H and E, we have
fi = (c2/£) V2D + (k/4w) V2 curl D.
For a plane wave which is propagated parallel to the axis of x,
this equation reduces to
k_
47T
k
47T ~& '
and, as MacCullagh had shown in 1836,f these equations are
competent to represent the rotation of the plane of polarization.
In the closing years of the nineteenth century, the general
theory of aether and electricity assumed a new form. But
before discussing the memoirs in which the new conception was
unfolded, we shall consider the progress which had been made
since the middle of the century in the study of conduction in
liquid and gaseous media.
*Cf. p. 186. tcf. p. 175.
2B2
( 372 )
CHAPTEE XI.
CONDUCTION IN SOLUTIONS AND GASES, FROM FARADAY TO
J. J. THOMSON.
THE hypothesis which Grothuss and Davy had advanced* to
explain the decomposition of electrolytes was open to serious
objection in more than one respect. Since the electric force
was supposed first to dissociate the molecules of the electrolyte
into ions, and afterwards to set them in motion toward the
electrodes, it would seem reasonable to expect that doubling
the electric force would double both the dissociation of the
molecules and the velocity of the ions, and would therefore
quadruple the electrolysis — an inference which is not verified
by observation. Moreover it might be expected, on Grothuss'
theory, that some definite magnitude of electromotive force
would be requisite for the dissociation, and that no electrolysis
at all would take place when the electromotive force was below
this value, which again is contrary to experience.
A way of escape from these difficulties was first indicated, in
1850, by Alex. Williamson,-)- who suggested that in compound
liquids decompositions and recombinations of the molecules are
continually taking place throughout the whole mass of the liquid,
quite independently of the application of an external electric
force. An atom of one element in the compound is thus paired
now with one and now with another atom of another element,
and in the intervals between these alliances the atom may be
regarded as entirely free. In 1857 this idea was made by
* Cf. p. 78.
f Phil. Mag. xxxvii (1850), p. 350 ; Liebig's Annulen d. Chem. u. Pharni.
Ixxvii (1851) p. 37.
Conduction in Solutions and Gase*, etc. 373
K. Clausius,* of Zurich, the basis of a theory of electrolysis.
According to it, the electromotive force emanating from the
electrodes does not effect the dissociation of the electrolyte
into ions, since a degree of dissociation sufficient for the purpose
already exists in consequence of the perpetual mutability of the
molecules of the electrolyte. Clausius assumed that these ions
are in opposite electric conditions; the applied electric force
therefore causes a general drift of all the ions of one kind
towards the anode, and of all the ions of the other kind towards
the cathode. These opposite motions of the two kinds of ions
constitute the galvanic current in the liquid.
The merits of the Williamson-Clausius hypothesis were not
fully recognized for many years ; but it became the foundation
of that theory of electrolysis which was generally accepted at
the end of the century.
Meanwhile another aspect of electrolysis was receiving
attention. It had long been known that the passage of a
current through an electrolytic solution is attended not only
by the appearance of the products of decomposition at the
electrodes, but also by changes of relative strength in different
parts of the solution itself. Thus in the electrolysis of a solution
of copper sulphate, with copper electrodes, in which copper is
dissolved off the anode and deposited on the cathode, it is found
that the concentration of the solution diminishes near the
cathode, and increases near the anode. Some experiments on
the subject were made by Faradayf in 1835 ; and in 1844 it
was further investigated by Frederic Daniell and W. A. Miller, J
who explained it by asserting that the cation and anion have
not (as had previously been supposed) the same facility of
moving to their respective electrodes ; but that in many cases
the cation appears to move but little, while the transport is
effected chiefly by the anion.
* Ann. d. Phys. ci (1857), p. 338 ; Phil. Mag. xv (1858), p. 94.
t Exper. Res. §§ 525-53C.
* Phil. Trans., 1844, p. 1. Cf. also Pouillet, Comptes Rendus xx (1845),
p. 1544.
374 Conduction in Solutions and Gases,
This idea was adopted by W. Hittorf, of Minister, who, in the
years 1853 to 1859, published* a series of memoirs on the
migration of the ions. Let the velocity of the anions in the
solution be to the velocity of the cations in the ratio v : u.
Then it is easily seen that if (u + v) molecules of the electrolyte
are decomposed by the current, and yielded up as ions at the
electrodes, v of these molecules will have been taken from the
fluid on the side of the cathode, and u of them from the fluid
on the side of the anode. By measuring the concentration of
the liquid round the electrodes after the passage of a current,
Hittorf determined the ratio v/u in a large number of cases of
electrolysis.!
The theory of ionic movements was advanced a further
stage by F. W. KohlrauschJ (I. 1840, d. 1910), of Wurzburg.
Kohlrausch showed that although the ohmic specific conduc-
tivity k of a solution diminishes indefinitely as the strength
of the solution is reduced, yet the ratio k/m, where m denotes
the number of gramme-equivalents§ of salt per unit volume, tends
to a definite limit, when the solution is indefinitely dilute. This
limiting value may be denoted by A. He further showed that
A may be expressed as the sum of two parts, one of which
depends on the cation, but is independent of the nature of the
anion; while the other depends on the anion, but not on the
cation — a fact which may be explained by supposing that, in
very dilute solutions, the twos ions move independently under
the influence of the electric force. Let u and v denote the
velocities of the cation and anion respectively, when the
potential difference per cm. in the solution is unity : then the
total current carried through a cube of unit volume is mE(u + v),
where E denotes the electric charge carried by one gramme-
# Ann. d. Phys. Ixxxix (1853), p. 177 ; xcviii (1856), p. 1 ; ciii (1858), p. 1 ;
cvi (1859), pp. 337, 513.
t The ratio v/(u + v) was termed by Hittorf the transport, number of the anion.
J Ann d. Phys. vi (1879), pp. 1, 145. The chief results had been communicated
to the Academy of Gottingen in 1876 and 1877.
§ A gramme-equivalent means a muss of the salt whose weight in grammes
is the molecular weight divided by the valency of the ions.
from Faraday to J. J. Thomson. 375
equivalent of ion.* Thus mE (u + v) = total current = k = raA,
or A = E (u + v). The determination of v/u by the method of
Hittorf, and of (u + v) by the method of Kohlrausch, made it
possible to calculate the absolute velocities of drift of the ions
from experimental data.
Meanwhile, important advances in voltaic theory were
being effected in connexion with a different class of investi-
gations.
Suppose that two mercury electrodes are placed in a solution
of acidulated water, and that a difference of potential, insufficient
to produce continuous decomposition of the water, is set up
between the electrodes by an external agency. Initially a
slight electric current — the polarizing current,f as it is called —
is observed; but after a short time it ceases; and after its cessation
the state of the system is one of electrical equilibrium. It is
evident that the polarizing current must in some way have set
up in the cell an electromotive force equal and opposite to the
external difference of potential ; and it is also evident that the
seat of this electromotive force must be at the electrodes, which
are now said to be polarized.
An abrupt fall of electric potential at an interface between
two media, such as the mercury and the solution in the present
case, requires that there should be a field of electric force, of
considerable intensity, within a thin stratum at the interface >
and this must owe its existence to the presence of electric
charges. Since there is no electric field outside the thin stratum,
there must be as much vitreous as resinous electricity present ;
but the vitreous charges must preponderate on one side of the
stratum, and the resinous charges on the other side ; so that
the system as a whole resembles the two coatings of a con-
denser with the intervening dielectric. In the case of the
* i.e. E is 96580 coulombs.
t The phenomenon of voltaic polarization was discovered by Hitter in 1803.
Hitter explained it by comparing the action of the polarizing current to that of a
current which is used to charge a condenser. Volta in 1805 put forward the
alternative explanation, that the products of decomposition set tip a reverse
electromotive force.
376 Conduction in Solutions and Gases,
polarized mercury cathode in acidulated water, there must be
on the electrode itself a negative charge : the surface of this
electrode in the polarized state may be supposed to be either
mercury, or mercury covered with a layer of hydrogen. In
the solution adjacent to the electrode, there must be an excess
of cations and a deficiency of anions, so as to constitute the
other layer of the condenser : these cations may be either
mercury cations dissolved from the electrode, or the hydrogen
cations of the'solution.
It was shown in 1870 by Cromwell Fleetwood Varley* that
a mercury cathode, thus polarized in acidulated water, shows a
tendency to adopt a definite superficial form, as if the surface-
tension at the interface between the mercury and the solution
were in some way dependent on the electric conditions. The
matter was more fully investigated in 1873 by a young
French physicist, then preparing for his inaugural thesis,
Gabriel Lippmann.f In Lippmann's instrumental disposition,
which is called a capillary electrometer, mercury electrodes are
immersed in acidulated water : the anode HQ has a large
surface, wkile^the cathode H has a variable surface S small in
comparison. When the external electromotive force is applied,
it is easily seen that the fall of potential at the large electrode
is only slightly affected, while the fall of potential at the small
electrode is altered by polarization by an amount practically
equal to the external electromotive force. Lippmann found
that the constant of capillarity of the interface at the small
electrode was a function of the external electromotive force, and
therefore of the difference of potential between the mercury
and the electrolyte.
Let V denote the external electromotive force: we may,
without loss of generality, assume the potential of £[„ to be zero,
so that the potential of H is - V. The state of the system may
be varied by altering either V or /S; we assume that these
* Phil. Trans, clxi (1871), p. 129.
f Comptes Rendus Ixxvi (1873), p. 1407. Phil. Mag. xlvii (1874), p. 281.
Ann. de Chim. et de Phys. v (1875), p. 494, xii (1877), p. 265.
from Faraday to J . J . Thomson. 377
alterations may be performed independently, reversibly, and
isothermally, and that the state of the large electrode H,} is not
altered thereby. Let de denote the quantity of electricity which
passes through the cell from 5"0 to H, when the state of the
system is thus varied : then if E denote the available energy of
the system, and y the surface-tension at H, we have
dE = ydS + Vde,
y being measured by the work required to increase the surface
when no electricity flows through the circuit.
In order that equilibrium may be re-established between the
electrode and the solution when the fall of potential at the
cathode is altered, it will be necessary not only that some
hydrogen cations should come out of the solution and be
deposited on the electrode, yielding up their charges, but also
that there should be changes in the clustering of the charged
ions of hydrogen, mercury, and sulphion in the layer of the
solution immediately adjacent to the electrode. Each of these
circumstances necessitates a flow of electricity in the outer
circuit : in the one case to neutralize the charges of the cations
deposited, and in the other case to increase the surface-density
of electric charge on the electrode, which forms the opposite
sheet of the quasi-condenser. Let Sf (V) denote the total
quantity of electricity which has thus flowed in the circuit
when the external electromotive force has attained the value V.
Then evidently
so
dE= {y+ Vf(V)\dS + VSf (V}dV.
Since this expression must be an exact differential, we have
so that - dy/d V is equal to that flux of electricity per unit of
new surface formed, which will maintain the surface in a
378 Conduction in Solutions and Gases,
constant condition (V being constant) when it is extended.
Integrating the previous equation, we have
Lippmann found that when the external electromotive force
was applied, the surface-tension increased at first, until, when
the external electromotive force amounted to about one volt,
the surface-tension attained a maximum value, after which it
diminished. He found that d-y/d F2 was sensibly independent
of F, so that the curve which represents the relation between
7 and F is a parabola.*
The theory so far is more or less independent of assumptions
as to what actually takes place at the electrode : on this latter
question many conflicting views have been put forward. In
1878 Josiah Willard Gibbs,t of Yale (b. 1839, d. 1903), discussed
the problem on the supposition that the polarizing current is
simply an ordinary electrolytic conduction-current, which
causes a liberation of hydrogen from the ionic form at the
cathode. If this be so, the amount of electricity which passes
through the cell in any displacement must be proportional to
the quantity of hydrogen which is yielded up to the electrode
in the displacement; so that dy/dV must be proportional to
the amount of hydrogen deposited per unit area of the
electrode.:}:
A different view of the physical conditions at the polarized
electrode was taken by Helmholtz,§ who assumed that the ions
of hydrogen which are brought to the cathode by the polarizing
current do not give up their charges there, but remain in the
vicinity of the electrode, and form one face of a quasi-condenser
* Lippman, Coniptes Eendus, xcv (1882), p. 686.
t Trans. Conn. Acad. iii (1876-1878), pp. 108, 343; Gibbs' Scientific Papers,
i, p. 55.
J This is embodied in equation (690) of Gibbs' memoir.
§ Berlin Monatsber., 1881, p. 945 ; Wiss. Abh. i, p. 925 ; Ann. d. Phys. xvi.
(1882), p. 31. Cf. also Planck, Ann. d. Phys. xliv (1891), p. 385.
from Faraday to J . J. Thomson. 379
of which the other face is the electrode itself.* If a denote
the surface-density of electricity on either face of this quasi-
condenser, we have, therefore,
de = - d(Sa) ; so a = dyfd V.
This equation shows that when dyldV is zero — i.e., when
the surface-tension is a maximum — a must be zero ; that is to
say, there must be no difference of potential between the
mercury and the electrolyte. The external electromotive force
is then balanced entirely by the discontinuity of potential at
the other electrode J7"0 ; and thus a method is suggested of
measuring the latter discontinuity of potential. All previous
measurements of differences of potential had involved the
employment of more than one interface ; and it was not known
how the measured difference of potential should be distributed
among these interfaces ; so that the suggestion of a means of
measuring single differences of potential was a distinct advance,
even though the hypotheses on which the method was based
were somewhat insecure.
A further consequence deduced by Helmholtz from this
theory leads to a second method of determining the difference
of potential between mercury and an electrolyte. If a mercury
surface is rapidly extending, and electricity is not rapidly
transferred through the electrolyte, the electric surface-density
in the double layer must rapidly decrease, since the same
quantity of electricity is being distributed over an increasing
area. Thus it may be inferred that a rapidly extending
mercury-surface in an electrolyte is at the same potential as
the electrolyte.
This conception is realized in the dropping-electrode, in
* The conception of double layers of electricity at the surface of separation of
two bodies had been already applied by Helmholtz to explain various other
phenomena — e.g., the Volta contact-difference of potential of two metals, fiictional
electricity, and *' electric endosmose," or the transport of fluid which occurs when
an electric current is passed through two conducting liquids separated by a porous
barrier. Cf. Helmholtz, Berlin Monatsberichte, February 27, 1879 ; -Ann. d. Phys.
vii (1879), p. 337 ; Helmholtz, Wiss. Abh. i, p. 855.
380 Conduction in Solutions and Gases ,
which a jet of mercury, falling from a reservoir into an electro-
lytic solution, is so adjusted that it breaks into drops when
the jet touches the solution. According to Helmholtz's
conclusion there is no difference of potential between the
drops and the electrolyte ; and therefore the difference of
potential between the electrolyte and a layer of mercury
underlying it in the same vessel is equal to the difference of
potential between this layer of mercury and the mercury
in the upper reservoir, which difference is a measurable
quantity.
It will be seen that according to the theories both of Gibbs
and of Helmholtz, and indeed according to all other theories on
the subject,* d^ldV is zero for an electrode whose surface is
* E.g., that of Warburg, Ann. d. Phys. xli (1890), p. 1. In this it is assumed
that the electrolytic solution near the electrodes originally contains a salt of
mercury in solution. When the external electromotive force is applied, a conduc-
tion-current passes through the electrolyte, which in the hody of the electrolyte
is carried by the acid and hydrogen ions. Warburg supposed that at the
cathode the hydrogen ions react with the salt of mercury, reducing it to metallic
mercury, which is deposited on the electrode. Thus a considerable change in
concentration of the salt of mercury is caused at the cathode. At the anode, the
acid ions carrying the current attack the mercury of the electrode, and thus
increase the local concentration of the mercuric salt ; but on account of the size of
the anode this increase is trivial and may be neglected.
Warburg thus supposed that the electromotive force of the polarized cell is
really that of a concentration cell, depending on the different concentrations of
mercuric salt at the electrodes. He found dy/dV to be equal to the amount of
mercuric salt at the cathode per unit area of cathode, divided by the electro-
chemical equivalent of mercury. The equation previously obtained is thus
presented in a new physical interpretation.
Warburg connected the increase of the surface-tension with the fact that the
surface-tension between mercury and a solution always increases when the con-
centration of the solution is diminished. His theory, of course, leads to no
conclusion regarding the absolute potential difference between the mercury and the
solution, as Helmholtz' does.
Alan electrode whose surface is rapidly increasing — e.g., a dropping electrode —
Warburg supposed that the surface-density of mercuric salt tends to zero, so
dyldV is zero.
The explanation of dropping electrodes favoured by Nernst, Beilage zu den
Ann. d. Phys. Iviii (1896), is that the difference of potential corresponding to the
equilibrium between the mercury and the electrolyte is instantaneously
established ; but that ions are withdrawn from the solution in order to form the
double layer necessary for this, and that these ions are carried down with the drops
from Faraday to J. J . Thomson. 381
rapidly increasing — e.g., a dropping electrode; that is to say,
the difference of potential between an ordinary mercury
electrode and the electrolyte, when the surface-tension has its
maximum value, is equal to the difference of potential between
a dropping-electrode and the same electrolyte. This result has
been experimentally verified by various investigators, who have
shown that the applied electromotive force when the surface-
tension has its maximum value in the capillary electrometer, is
equal to the electromotive force of a cell having as electrodes a
large mercury electrode and a dropping electrode.
Another memoir which belongs to the same period of
Helmholtz' career, and which has led to important develop-
ments, was concerned with a special class of voltaic cells. The
most usual type of cell is that in which the positive electrode
is composed of a different metal from the negative electrode,
and the evolution of energy depends on the difference in the
chemical affinities of these metals for the liquids in the cell.
But in the class of cells now considered* by Helmholtz, the
two electrodes are composed of the same metal (say, copper) ;
and the liquid (say, solution of copper sulphate) is more con-
centrated in the neighbourhood of one electrode than in the
neighbourhood of the other. When the cell is in operation, the
salt passes from the places of high concentration to the places
of low concentration, so as to equalize its distribution ; and this
process is accompanied by the flow of a current in the outer
circuit between the electrodes. Such cells had been studied
experimentally by James Moser a short time previously! to
Helmholtz' investigation.
The activity of the cell is due to the fact that the available
energy of a solution depends on its concentration ; the molecules
of mercury, until the upper layer of the solution is so much impoverished that the
double layer can no longer be formed. The impoverishment of the upper layer of
the solution has actually been observed by Palniaer, Zeitsch. Phys. Chem. xxv
(1898), p. 265 ; xxviii (1899), p. 257 ; xxxvi (1901), p. 664.
* Berlin Monatsber., 1877, p. 713 ; Phil. Mag. (5) v (1878), p. 348; reprinted
with additions in Ann. d. Phys. iii (1878), p. 201.
t Ann. d. Phys. iii (1878), p. 216.
382 Conduction in Solutions and Gases ,
of salt, in passing from a high to a low concentration, are
therefore capable of supplying energy, just as a compressed gas
is capable of supplying energy when its degree of compression
is reduced. To examine the matter quantitatively, let nf(nf V)
denote the term in the available energy of a solution, which is
due to the dissolution of n gramme-molecules of salt in a volume
V of pure solvent ; the function / will of course depend also on
the temperature. Then when dn gramme-molecules of solvent
are evaporated from the solution, the decrease in the available
energy of the system is evidently equal to the available energy of
dn gramme-molecules of liquid solvent, less the available energy
of dn gramme-molecules of the vapour of the solvent, together
with nf(n/ V) less nf{n/(V-v dn) } , where v denotes the volume
of one gramme-molecule of the liquid. But this decrease in
available energy must be equal to the mechanical work supplied
to the external world, which is dn . p± (v - v), if pl denote the
vapour-pressure of the solution at the temperature in question,
and v denote the volume of one gramme-molecule of vapour.
We have therefore
dn . pi (v' - v) = — available energy of dn gramme-molecules of
solvent vapour
+ available energy of dn gramme-molecules of
liquid solvent
+ nf(n/ V) - nf {n/( V-v dn) \ .
Subtracting from this the equation obtained by making n zero,
we have
dn . (Pi - p0) (v - v) = nf(n/ V) - nf( n/( V - v dn) } ,
where pQ denotes the vapour-pressure of the pure solvent at the
temperature in question ; so that
(Pi -Po) <>' - v) = - (n'/V*)f(n/V)v.
Now, it is known that when a salt is dissolved in water, the
vapour-pressure is lowered in proportion to the concentration
of the salt — at any rate when the concentration is small : in
from Faraday to J . J. Thomson. 383
fact, by the law of Kaoult, (p0-pi)/po is approximately equal to
nv/ V ; so that the previous equation becomes
p. V(v' -f.) -*/(»/ F).
Neglecting v in comparison with v', and making use of the
equation of state of perfect gases (namely,
pjt = ST.
where T denotes the absolute temperature, and R denotes the
constant of the equation of state), we have
and therefore
Thus in the available energy of one gramme-molecule of a
dissolved salt, the term which depends on the concentration is
proportional to the logarithm of the concentration ; and hence,
if in a concentration-cell one gramme-molecule of the salt
passes from a high concentration c2 at one electrode to a low
concentration GI at the other electrode, its available energy is
thereby diminished by an amount proportional to log (c2/c,).
The energy which thus disappears is given up by the system in
the form of electrical work; and therefore the electromotive
force of the concentration-cell must be proportional to log (Cz/cJ..
The theory of solutions and their vapour-pressure was
not at the time sufficiently developed to enable Helmholtz
to determine precisely the coefficient of log (c2/Ci) in the
expression.*
An important advance in the theory of solutions was effected
in 1887, by a young Swedish physicist, Svante Arrhenius.f
* The formula given by Helmholtz was that the electromotive force of the cell
is equal to b(l - ri) v log (czjc\), where ci and c\ denote the concentrations of the solu-
tion at the electrodes, v denotes the volume of one gramme of vapour in equilibrium
with the water at the temperature in question, n denotes the transport number for
the cation (Hittorfs 1/w), and b denotes q x the lowering of vapour- pressure when
one gramme-equivalent of salt is dissolved in q grammes of water, where q denotes
a large number.
t Zeitschrift fur phys. Chem. i (1887), p. 631. Previous investigations, in
which the theory was to some extent foreshadowed, were published in Bihang
till Svenska Vet. Ak. Forh. viii (1884), Nos. 13 and 14.
384 Conduction in Solutions and Gases,
Interpreting the properties discovered by Kohlrausch* in the
light of the ideas of Williamson and Clausius regarding the
spontaneous dissociation of electrolytes, Arrhenius inferred that
in very dilute solutions the electrolyte is completely dissociated
into ions, but that in more concentrated solutions the salt is
less completely dissociated; and that as in all solutions the
transport of electricity in the solution is effected solely by the
movement of ions, the equivalent conductivityf must be pro-
portional to the fraction which expresses the degree of ionization.
By aid of these conceptions it became possible to estimate the
dissociation quantitatively, and to construct a general theory
of electrolytes.
Contemporary physicists and chemists found it difficult
at first to believe that a salt exists in dilute solution only
in the form of ions, e.g. that the sodium and chlorine exist
separately and independently in a solution of common salt.
But there is a certain amount of chemical evidence in favour
of Arrhenius' conception. For instance, the tests in chemical
analysis are really tests for the ions ; iron in the form of a fer-
rocyanide, and chlorine in the form of a chlorate, do not respond
to the characteristic tests for iron and chlorine respectively,
which are really the tests for the iron and chlorine ions.
The general acceptance of Arrhenius' views was hastened
by the advocacy of Ostwald, who brought to light further
evidence in their favour. For instance, all permanganates
in dilute solution show the same purple colour; and
Ostwald considered their absorption-spectra to be identical ;J
this identity is easily accounted for on Arrhenius' theory, by
supposing that the spectrum in question is that of the anion
which corresponds to the acid radicle. The blue colour
which is observed in dilute solutions of copper salts, even
when the strong solution is not blue, may in the same way be
* Cf. p. 374.
t I.e. the ohmic specific conductivity of the solution divided by the number of
gramme-equivalents of salt per unit volume.
J Examination of the spectra with higher dispersion does not altogether
confirm this conclusion.
from Faraday to J .J. Thomson. 385
ascribed to a blue copper cation. A striking instance of the
same kind is afforded by ferric sulphocyanide ; here the strong
solution shows a deep red colour, due to the salt itself ; but on
dilution the colour disappears, the ions being colourless.
If it be granted that ions can have any kind of permanent
existence in a salt solution, it may be shown from thermo-
dynamical considerations that the degree of dissociation must
increase as the dilution increases, and that at infinite dilution
there must be complete dissociation. For the available energy
of a dilute solution of volume V, containing «j gramme-molecules
of one substance, >/2 gramme-molecules of another, and so on, is
(as may be shown by an obvious extension of the reasoning
already employed in connexion with concentration-cells)*
r (T) + RT^nr log (UT! V) + the available energy
possessed by the solvent before the introduction of the solutes,
where 0r (T) depends on T and on the nature of the rth solute,
but not on V, and R denotes the constant which occurs in the
equation of state of perfect gases. When the system is in
equilibrium, the proportions of the reacting substances will
be so adjusted that the available energy has a stationary
value for small virtual alterations Swj, &^, ...... of the
proportions ; and therefore
0 - SSnr .<t>r(T) + RT2$nr.log (nrjV)
Applying this to the case of an electrolyte in which the
disappearance of one molecule of salt (indicated by the suffix ,)
gives rise to one cation (indicated by the suffix 2) and one anion
(indicated by the suffix 3), we have B^ = - £7^ = - Sn* ; so the
equation becomes
0 = 0, (T) - 02 (T) - 03 (T) + RT log (n, V/n.n,) - RT,
or
= a function of T only.
* Cf. pp. 382-383.
2 C
386 Conduction in Solutions and Gases,
Since in a neutral solution the number of anions is equal to the
number of cations, this equation may be written
nf = Fw-i x a function of T only ;
it shows that when V is very large (so that the solution is very
dilute), n2 is very large compared with n^ ; that is to say, the
salt tends towards a state of complete dissociation.
The ideas of Arrhenius contributed to the success of Walther
Nernst* in perfecting Helmholtz' theory of concentration-cells,
and representing their mechanism in a much more definite
fashion than had been done heretofore.
In an electrolytic solution let the drift-velocity of the
cations under unit electric force be u, and that of the anions
be vt so that the fraction uj(u + v} of the current is transported
by the cations, and the fraction v/(u + v) by the anions. If the
concentration of the solution be Cj at one electrode, and c2 at the
other, it follows from the formula previously found for the
available energy that one gramme - ion of cations, in moving
from one electrode to the other, is capable of yielding up an
amountf RT log (c2/c,) of energy; while one gramme - ion
of anions going in the opposite direction must absorb the same
amount of energy. The total quantity of work furnished when
one gramme-molecule of salt is transferred from concentration
ct to concentration c{ is therefore
u + v
The quantity of electric charge which passes in the circuit
when one gramme-molecule of the salt is transferred is pro-
portional to the valency v of the ions, and the work furnished
is proportional to the product of this charge and the electro-
*Zeitschr. fur phys. Chem. ii (1888), p. 613; iv (1889), p. 129; Berlin
Sitzungsberichte, 1889, p. 83 ; Ann. d. Phys. xlv (1892), p. 360. Cf. also
Max Planck, Ann. d. Phys. xxxix (1890), p. 161 ; xl (1890), p. 561.
t The correct law of dependence of the available energy on the temperature was
by this time known.
from Faraday to J . J . Thomson. 387
motive force E of the cell ; so that in suitable units we have
-, RTu-v. c,
E = -- - log -.
v u + v Ci
A typical concentration-cell to which this formula may be
applied may be constituted in the following way : — Let a
quantity of zinc amalgam, in which the concentration of zinc
is d, be in contact with a dilute solution of zinc sulphate, and
let this in turn be in contact with a quantity of zinc amalgam
of concentration cz. When the two masses of amalgam are con-
nected by a conducting wire outside the cell, an electric current
flows in the wire from the weak to the strong amalgam,* while
zinc cations pass through the solution from the strong amalgam
to the weak. The electromotive force of such a cell, in which
the current may be supposed to be carried solely by cations, is
RT. c,
— lo-
V °
Not content with the derivation of the electromotive force
from considerations of energy, Nernst proceeded to supply a
definite mechanical conception of the process of conduction in
electrolytes. The ions are impelled by the electric force asso-
ciated with the gradient of potential in the electrolyte. But
this is not the only force which acts on them ; for, since their
available energy decreases as the concentration decreases, there
must be a force assisting every process by which the concentra-
tion is decreased. The matter may be illustrated by the analogy
of a gas compressed in a cylinder fitted with a piston; the
available energy of the gas decreases as its degree of compression
decreases; and therefore that movement of the piston which
tends to decrease the compression is assisted by a force — the
"pressure" of the gas on the piston. Similarly, if a solution
were contained within a cylinder fitted with a piston which is
permeable to the pure solvent but not to the solute, and if the
whole were immersed in pure solvent, the available energy of
* It will hardly be necessary to remark that this supposed direction of the
.current is purely conventional.
2 C 2
388 Conduction in Solutions and Gases,
the system would be decreased if the piston were to move
outwards so as to admit more solvent into the solution; and
therefore this movement of the piston would be assisted by a
force — the "osmotic pressure of the solution," as it is called.*
Consider, then, the case of a single electrolyte supposed to
be perfectly dissociated ; its state will be supposed to be the
same at all points of any plane at right angles to the axis of x.
Let v denote the valency of the ions, and V the electric potential
at any point. Sincef the available energy of a given quantity of
a substance in very dilute solution depends on the concentration
in exactly the same way as the available energy of a given
quantity of a perfect gas depends on its density, it follows that
the osmotic pressure p for each ion is determined in terms of
the concentration and temperature by the equation of state
of perfect gases
Mp = ETc,
where M denotes the molecular weight of the salt, and c the
mass of salt per unit volume.
Consider the cations contained in a parallelepiped at the
place x, whose cross-section is of unit area and whose length
is dx. The mechanical force acting on them due to the electric
field is - (vc/M) d Vfdx . dx, and the mechanical force on them
due to the osmotic pressure is - dp/dx . dx. If u denote the
velocity of drift of the cations in a field of unit electric force,
the total amount of charge which would be transferred by
cations across unit area in unit time under the influence of the
electric forces alone would be - (uvc/M) d V/dx ; so, under the
influence of both forces, it is
_uvcidV_ ET dc\
M\dx cv dx)
Similarly, if v denote the velocity of drift of the anions in a
* Cf . van't Hoff, Svenska Vet.-Ak. Handlingar xxi (1886), No. 17; Zeitschrift
fiir Phys. Chem. i (1887), p. 481.
t As follows from the expression obtained, supra, p. 383.
from Faraday to jf . J . Thomson. 389
unit electric field, the charge transferred across unit area in
unit time by the anions is
vvcfdV^ RT dc\
M\dx cv dx)
We have therefore, if the total current be denoted by i,
. vc dV RT do
-u+M^-u-v>^Tdx>
or
dV 7 Mdx u-v RT dc ,
- -T- dx = - -— ^ + -- — dx.
dx (u + v)vc u + v vc dx
The first term on the right evidently represents the product of
the current into the ohmic resistance of the parallelepiped dx,
while the second term represents the internal electromotive
force of the parallelepiped. It follows that if r denote the
specific resistance, we must have
u + v = Mjrvc,
in agreement with Kohlrausch's equation ;* while by integrating
the expression for the internal electromotive force of the
parallelepiped dx, we obtain for the electromotive force of a
cell whose activity depends on the transference of electrolyte
between the concentrations c, and cz, the value
u-v RT fl dc .
-- - T- <te>
u + v v c dx
u-v RT , c,
or — log-,
u + v v GI
in agreement with the result already obtained.
It may be remarked that although the current arising from
a concentration cell which is kept at a constant temperature is
capable of performing work, yet this work is provided, not by
any diminution in the total internal energy of the cell, but by
the abstraction of thermal energy from neighbouring bodies.
This indeed (as may be seen by reference to W. Thomson's general
* Cf. P. 374.
390 Conduction in Solutions and Gases,
equation of available energy)* must be the case with any
system whose available energy is exactly proportional to the
absolute temperature.
The advances which were effected in the last quarter of the
nineteenth century in regard to the conduction of electricity
through liquids, considerable though these advances were, may
be regarded as the natural development of a theory which had
long been before the world. It was otherwise with the kindred
problem of the conduction of electricity through gases : for
although many generations of philosophers had studied the
remarkable effects which are presented by the passage of a
current through a rarefied gas, it was not until recent times
that a satisfactory theory of the phenomena was discovered.
Some of the electricians of the earlier part of the eighteenth
century performed experiments in vacuous spaces ; in particular,
Hauksbeef in 1705 observed a luminosity when glass is rubbed
in rarefied air. But the first investigator of the continuous
discharge through a Tarefied gas seems to have been Watson,!
who, by means of an electrical machine, sent a current through
an exhausted glass tube three feet long and three inches in
diameter. " It was," he wrote, " a most delightful spectacle,
when the room was darkened, to see the electricity in its
passage : to be able to observe not, as in the open air, its
brushes or pencils of rays an inch or two in length, but here
the coruscations were of the whole length of the tube between
the plates, that is to say, thirty-two inches." Its appearance
he described as being on different occasions " of a bright silver
hue," " resembling very much the most lively coruscations of
the aurora borealis," and " forming a continued arch of lambent
flame." His theoretical explanation was that the electricity " is
seen, without any preternatural force, pushing itself on through
the vacuum by its own elasticity, in order to maintain the
* Cf. p. 241.
t Phil. Trans, xxiv (1705), p. 2165. Fra. Hauksbee, Physico- Mechanical
Experiments, London, 1709.
I Phil. Trans, xlv (1748), p. 93, xlvii (1752), p. 362.
from Faraday to J . J . Thomson. 391
equilibrium in the machine " — a conception which follows
naturally from the combination of Watson's one-fluid theory
with the prevalent doctrine of electrical atmospheres.*
A different explanation was put forward by Nollet, who
performed electrical experiments in rarefied air at about the
same time as Watson,f and saw in them a striking confirmation
of his own hypothesis of efflux and afflux of electric matter.J
According to Nollet, the particles of the effluent stream collide
with those of the affluent stream which is moving in the
opposite direction ; and being thus violently shaken, are excited
to the point of emitting light.
Almost a century elapsed before anything more was dis-
covered regarding the discharge in vacuous spaces. But in
1838 Faraday, § while passing a current from the electrical
machine between two brass rods in rarefied air, noticed that
the purple haze or stream of light which proceeded from the
positive pole stopped short before it arrived at the negative
rod. The negative rod, which was itself covered with a con-
tinuous glow, was thus separated from the purple column by
a narrow dark space: to this, in honour of its discoverer,
the name Faraday's dark space has generally been given by
subsequent writers.
That vitreous and resinous electricity give rise to different
types of discharge had long been known; and indeed, as we
have seen,) | it was the study of these differences that led
Franklin to identify the electricity of glass with the superfluity of
fluid, and the electricity of amber with the deficiency of it. But
phenomena of this class are in general much more complex
than might be supposed from the appearance which they
present at a first examination ; and the value of Faraday's
discovery of the negative glow and dark space lay chiefly in
the simple and definite character of these features of the
discharge, which indicated them as promising subjects for
further research. Faraday himself felt the importance of
* Cf. ch. ii. f Nollet, Recherches sur FElectricite, 1749, troisiemediscours.
t Cf. p. 40. § Phil. Trans., 1838 ; Exper. Res. i, § 1526. || Cf. p. 44.
392 Conduction in Solutions and Gases,
investigations in this direction. " The results connected with
the different conditions of positive and negative discharge," he
wrote,* " will have a far greater influence on the philosophy of
electrical science than we at present imagine."
Twenty more years, however, passed before another notable
advance was made. That a subject so full of promise should
progress so slowly may appear strange ; but one reason at any
rate is to be found in the incapacity of the air-pumps then in
use to rarefy gases to the degree required for effective study
of the negative glow. The invention of Geissler's mercurial
air-pump in 1855 did much to remove this difficulty; and it
was in Geissler's exhausted tubes that Julius Plticker,t of Bonn,
studied the discharge three years later.
It had been shown by Sir Humphrey Davy in 1821 J that
one form of electric discharge — namely, the arc between carbon
poles — is deflected when a magnet is brought near to it.
Pliicker now performed a similar experiment with the vacuum
discharge, and observed a similar deflexion. But the most
interesting of his results were obtained by examining the
behaviour of the negative glow in the magnetic field ; when
the negative electrode was reduced to a single point, the whole
of the negative light became concentrated along the line of
magnetic force passing through this point. In other words,
the negative glow disposed itself as if it were constituted of
flexible chains of iron filings attached at one end to the
cathode.
Pliicker noticed that when the cathode was of platinum,
small particles were torn off it and deposited on the walls of
the glass bulb. " It is most natural," he wrote, " to imagine
that the magnetic light is formed by the incandescence of these
platinum particles as they are torn from the negative electrode."
He likewise observed that during the discharge the walls of
* Exper. Res., § 1523.
| Ann. d. Phys. ciii (1858), pp. 88, 151 ; civ (1858), pp. 113, 622 ; cv (1858),
p. 67; cvii (1859), p. 77. Phil. Mag. xvi (1858), pp. 119, 408; xviii (1859),
pp. 1, 7.
J Phil. Trans., 1821, p. 425.
from Faraday to J . J. Thomson. 393
the tube, near the cathode, glowed with a phosphorescent light,
and remarked that the position of this light was altered when
the magnetic field was changed. This led to another discovery ;
for in 1869 Plucker's pupil, W. Hittorf,* having placed a solid
body between a point-cathode and the phosphorescent light, was
surprised to find that a shadow was cast. He rightly inferred
from this that the negative glow is formed of rays which
proceed from the cathode in straight lines, and which cause the
phosphorescence when they strike the walls of the tube.
Hittorf's observation was amplified in 1876 by Eugen
Goldstein,f who found that distinct shadows were cast, not
only when the cathode was a single point, but also when it
formed an extended surface, provided the shadow-throwing
object was placed close to it. This clearly showed that the
cathode rays (a term now for the first time introduced) are not
emitted indiscriminately in all directions, but that each portion
of the cathode surface emits rays which are practically confined
to a single direction ; and Goldstein found this direction to be
normal to the surface. In this respect his discovery established
an important distinction between the manner in which cathode
rays are emitted from an electrode and that in which light is
emitted from an incandescent surface.
The question as to the nature of the cathode rays attracted
much attention during the next two decades. In the year
following Hittorf's investigation, Cromwell VarleyJ put forward
the hypothesis that the rays are composed of " attenuated par-
ticles of matter, projected from the negative pole by electricity" ;
and that it is in virtue of their negative charges that these
particles are influenced by a magnetic field. §
During some years following this, the properties of highly
* Ann. <1. Phys. cxxxvi (1869), pp. 1, 197; translated, Annales de Cbimie, xvi
(1869), p. 487.
t Berlin Monatsberichte, 1876, p. 279.
J Proc. Roy. Soc. xix (1871), p. 236.
§ Priestley in 1766 had shown that a current of electri6ed air flows from the
points of hodies which are electrified either vitreously or resinously : cf. Priestley's
History of Electricity, p. 591.
394 Conduction in Solutions and Gases,
rarefied gases were investigated by Sir William Crookes.
Influenced, doubtless, by the ideas which were developed in
connexion with his discovery of the radiometer, Crookes,* like
Varley, proposed to regard the cathode rays as a molecular
torrent : he supposed the molecules of the residual gas, coming
into contact with the cathode, to acquire from it a resinous charge,
and immediately to fly off normally to the surface, by reason of
the mutual repulsion exerted by similarly electrified bodies.
Carrying the exhaustion to a higher degree, Crookes was enabled
to study a dark space which under such circumstances appears
between the cathode and the cathode glow ; and to show that at
the highest rarefactions this dark space (which has since been gene-
rally known by his name) enlarges until the whole tube is occupied
by it. He suggested that the thickness of the dark space may
be a measure of the mean length of free path of the molecules.
" The extra velocity," he wrote, " with which the molecules
rebound from the excited negative pole keeps back the more
slowly moving molecules which are advancing towards that pole.
The conflict occurs at the boundary of the dark space, where the
luminous margin bears witness to the energy of the collisions."f
Thus according to Crookes the dark space is dark and the
glow bright because there are collisions in the latter and not in
the former. The fluorescence or phosphorescence on the walls
of the tube he attributed to the impact of the particles on
the glass.
Crookes spoke of the cathode rays as an " ultra-gaseous " or
" fourth state " of matter. These expressions have led some
later writers to ascribe to him the enunciation or prediction of
a hypothesis regarding the nature of the particles projected from
the cathode, which arose some years afterwards, and which we
shall presently describe ; but it is clear from Crookes' memoirs
that he conceived the particles of the cathode rays to be
ordinary gaseous molecules, carrying electric charges ; and by
* Phil. Trans, clxx (1879), pp. 135, 641 ; Phil. Mag. vii (1879), p. 57.
t Phil. Mag. vii (1879), p. 57.
from Faraday to J . J . 71wmson. 395
" a new state of matter " he understood simply a state in which
the free path is so long that collisions may be disregarded.
Crookes found that two adjacent pencils of cathode rays
appeared to repel each other. At the time this was regarded as
a direct confirmation of the hypothesis that the rays are streams
of electrically charged particles ; but it was shown later that
the deflexion of the rays must be assigned to causes other than
mutual repulsion.
How admirably the molecular- torrent theory accounts for
the deviation of the cathode rays by a magnetic field was shown
by the calculations of Eduard Riecke in 1881.* If the axis of
z be taken parallel to the magnetic force Ht the equations of
motion of a particle of mass ra, charge e, and velocity (u, v, w)
are
mdu/dt = evH, mdvjdt = - euH, mdw/dt = 0.
The last equation shows that the component of velocity of the
particle parallel to the magnetic force is constant; the other
equations give
u = A sin (eHt/m), v = A cos (eHi/m),
showing that the projection of the path on a plane at right
angles to the magnetic force is a circle. Thus, in a magnetic
field the particles of the molecular torrent describe spiral paths
whose axes are the lines of magnetic force.
But the hypothesis of Varley and Crookes was before long
involved in difficulties. Taitf in 1880 remarked that if the
particles are moving with great velocities, the periods of the
luminous vibrations received from them should be affected to a
measurable extent in accordance with Doppler's principle.
Tait tried to obtain this effect, but without success. It may,
however, be argued that if, as Crookes supposed, the particles
become luminous only when they have collided with other
particles, and have thereby lost part of their velocity, the
phenomenon in question is not to be expected.
* Gott. Nach., 2 February, 1881; reprinted, Ann. d. Phys. xiii (1881), p. 191.
t Proc. Roy. Soc. Edinb. x (1880). p. 430.
396 Conduction in Solutions and Gases ,
The alternative to the molecular-torrent theory is to suppose
that the cathode radiation is a disturbance of the aether. This
view was maintained by several physicists,* and notably by
Hertz,f who rejected Varley's hypothesis when he found
experimentally that the rays did not appear to produce any
external electric or magnetic force, and were apparently not
affected by an electrostatic field. It was, however, pointed
out by Fitz Gerald* that external space is probably screened
from the effects of the rays by other electric actions which
take place in the discharge tube.
It was further urged against the charged-particle theory
that cathode rays are capable of passing through films of metal
which are so thick as to be quite opaque to ordinary light ;§
it seemed inconceivable that particles of matter should not be
stopped by even the thinnest gold-leaf. At the time of Hertz's
experiments on the subject, an attempt to obviate this difficulty
was made by J.-J. Thomson,! | who suggested that the metallic
film when bombarded by the rays might itself acquire the
property of emitting charged particles, so that the rays which
were observed on the further side need not have passed through
the film. It was Thomson who ultimately found the true
explanation ; but this depended in part on another order of
ideas, whose introduction and development must now be
traced.
The tendency, which was now general, to abandon the
electron-theory of Weber in favour of Maxwell's theory
involved certain changes in the conceptions of electric charge.
* E.g. E. Wiedemann, Ann. d. Phys. x (1880), p. 202; translated, Phil. Mag.
x (1880), p. 357. E. Goldstein, Ann. d. Phys. xii (1881), p. 249.
t Ann. d. Phys. xix (1883), p. 782.
J Nature, November 5, 1896 ; Fitz Gerald's Scientific Writings, p. 433.
§ The penetrating power of the rays had been noticed by Hittorf, and by
E. "Wiedemann and Ebert, Sitzber. d. phys.-med. Soc. zu Erlangen, llth December,
1891. It was investigated more thoroughly by Hertz, Ann. d. Phys. xlv (1892),
p. 28, and by Philipp Lenard, of Bonn, Ann. d. Phys. li (1894), p. 225 ; lii (1894),
p. 23, who conducted a series of experiments on cathode rays which had passed out
of the discharge tube through a thin window of aluminium.
|| J. J. Thomson, Recent Researches, p. 126.
from Faraday to J . J . Thomson. 397
•
In the theory of Weber, electric phenomena were attributed
to the agency of stationary or moving charges, which could
most readily be pictured as having a discrete and atom-like
existence. The conception of displacement, on the other hand,
which lay at the root of the Maxwellian theory, was more in
harmony with the representation of electricity as something
of a continuous nature; and as Maxwell's views met with
increasing acceptance, the atomistic hypothesis seemed to have
entered on a period of decay. Its revival was due largely to the
advocacy of Helmholtz,* who, in a lecture delivered to the
Chemical Society of London in 1881, pointed outf that it was
thoroughly in accord with the ideas of Faraday,J on which
Maxwell's theory was founded. " If," he said, " we accept the
hypothesis that the elementary substances are composed of
atoms, we cannot avoid concluding that electricity also, positive
as well as negative, is divided into definite elementary portions
which behave like atoms of electricity."
When the conduction of electricity is considered in the
light of this hypothesis, it seems almost inevitable to conclude
that the process is of much the same character in gases as in
electrolytes ; and before long this view was actively maintained.
It had indeed long been known that a compound gas might be
decomposed by the electric discharge ; and that in some cases
the constituents are liberated at the electrodes in such a way
as to suggest an analogy with electrolysis. The question had
been studied in 1861 by Adolphe Perrot, who examined § the
gases liberated by the passage of the electric spark through
steam. He found that while the product of this action was
a detonating mixture of hydrogen and oxygen, there was a
decided preponderance of hydrogen at one pole and of oxygen
at the other.
The analogy of gaseous conduction to electrolysis was
applied by W. Giese,|| of Berlin, in 1882, in order to explain
* Cf. also G. Johnstone Stoney, Phil. Mag., May, 1881.
f Journ. Chem. Soc. xxxix (1881), p. 277. \ Cf. p. 200.
§ Annales de Chimie (3), Ixi, p. 161.
|| Ann. d. Phys. xvii (1882), pp. 1, 236, 519.
398 Conduction in Solutions and Gases ,
the conductivity of the hot gases of flames. " It is assumed,"
he wrote, " that in electrolytes, even before the application of
an external electromotive force, there are present atoms or
atomic groups — the ions, as they are called — which originate
when the molecules dissociate ; hy these the passage of electri-
city through the liquid is effected, for they are set in motion
by the electric field and carry their charges with them. We
shall now extend this hypothesis by assuming that in gases
also the property of conductivity is due to the presence of
ions. Such ions may be supposed to exist in small numbers
in all gases at the ordinary temperature and pressure ; and as
the temperature rises their numbers will increase."
Ideas similar to this were presented in a general theory of
the discharge in rarefied gases, which was devised two years
later by Arthur Schuster, of Manchester.* Schuster remarked
that when hot liquids are maintained at a high potential, the
vapours which rise from them are found to be entirely free
from electrification ; from which he inferred that a molecule
striking an electrified surface in its rapid motion cannot carry
away any part of the charge, and that one molecule cannot
communicate electricity to another in an encounter in which
both molecules remain intact. Thus he was led to the con-
clusion that dissociation of the gaseous molecules is necessary
for the passage of electricity through gases. f
Schuster advocated the charged-particle theory of cathode
rays, and by extending and interpreting an experiment of
Hittorf s was able to adduce strong evidence in its favour.
He placed the positive and negative electrodes so close to each
other that at very low pressures the Crookes' dark space
extended from the cathode to beyond the anode. In these
circumstances it was found that the discharge from the positive
electrode always passed to the nearest point of the inner
boundary of the Crookes' dark space — which, of course, was in
* Proc. Roy. Soc. xxxvii (1884), p. 317.
t In the case of an elementary gas, this would imply dissociation of the molecule
into two atoms chemically alike, but oppositely charged ; in electrolysis the
.dissociation is into two chemically unlike ions.
Jrom Faraday to J . J . Thomson* 399
the opposite direction to the ijathode. Thus, in the neighbour-
hood of the positive discharge, the current was flowing in two
opposite directions at closely adjoining places ; which could
scarcely happen unless the current in one direction were
carried by particles moving against the lines of force by
virtue of their inertia.
Continuing his researches, Schuster* showed in 1887 that
a steady electric current may be obtained in air between
electrodes whose difference of potential is but small, provided
that an independent current is maintained in the same
vessel ; that is to say, a continuous discharge produces in
the air such a condition that conduction occurs with the
smallest electromotive forces. This effect he explained by
aid of the hypothesis previously advanced ; the ions produced
by the main discharge become diffused throughout the vessel,
and, coming under the influence of the field set up by the
auxiliary electrodes, drift so as to carry a current between
the latter.
A discovery related to this was made in the same year by
Hertz,f in the course of the celebrated researches? which have
been already mentioned. Happening to notice that the passage
of one spark is facilitated by the passage of another spark in
its neighbourhood, he followed up the observation, and found
the phenomenon to be due to the agency of ultra-violet light
emitted by the latter spark. It appeared in fact that the
distance across which an electric spark can pass in air is
greatly increased when light of very short wave-length is
allowed to fall on the spark-gap. It was soon found§ that the
effective light is that which falls on the negative electrode
of the gap ; and Wilhelm Hallwachs|| extended the discovery
* Proc. Roy. Soc. xlii (1887), p. 371. Hittorf had discovered that very small
electromotive forces are sufficient to cause a discharge across a space through which
the cathode radiation is passing.
t Berlin Ber., 1887, p. 487 ; Ann. d. Phys. xxxi (1887), p. 983 ; Electric Waves
(English ed.), p. 63.
I Cf.p. 357.
§ By E. Wiedemann and Ebert, Ann. d. Phys. xxxiii (1888), p. 241.
|| Ann. d. Phys. xxxiii (1888), p. 301.
400 Conduction in Solutions and Gases,
by showing that when a sheet of metal is negatively electrified
and exposed to ultra-violet light, the adjacent air is thrown
into a state which permits the charge to leak rapidly away.
Interest was now thoroughly aroused in the problem of
conductivity in gases ; and it was generally felt that the best
hope of divining the nature of the process lay in studying the
discharge at high rarefactions. " If a first step towards under-
standing the relations between aether and ponderable matter is
to be made," said Lord Kelvin in 1893,* " it seems to me that
the most hopeful foundation for it is knowledge derived from
experiments on electricity in high vacuum."
Within the two following years considerable progress was
effected in this direction. J. J. Thomson,^ by a rotating-mirror
method, succeeded in measuring the velocity of the cathode rays,
finding it to be| 1*9 x 107 cm./sec. ; a value so much smaller than
that of the velocity of light that it was scarcely possible to
conceive of the rays as vibrations of the aether. A further
blow was dealt at the latter hypothesis when Jean Perrin,§
having received the rays in a metallic cylinder, found that
the cylinder became charged with resinous electricity. When
the rays were deviated by a magnet in such a way that they
could no longer enter the cylinder, it no longer acquired a
charge. This appeared to demonstrate that the rays transport
negative electricity.
With cathode rays is closely connected another type
of radiation, which was discovered in December, 1895, by
W. C. K6ntgen.ll The discovery seems to have originated
in an accident : a photographic plate which, protected in the
usual way, had been kept in a room in which vacuum-tube
experiments were carried on, was found on development to show
distinct markings. Experiments suggested by this showed
* Proc. Roy. Soc. liv (1893), p. 389.
t Phil. Mag. xxxviii (1894), p. 358.
£ The value found by the same investigator in 1897 was much larger than this.
$ Cornptes Rendus, cxxi (1895), p. 1130.
|| Sitzungsber. der Wiirzburger Physikal. -Medic. Gesellschaft, 1895 ; reprinted,
Ann. d. Phys. Ixiv (1898), pp. 1, 12; translated, Nature, liii (1896), p. 274.
Jrom Faraday to J. J . Thomson. 401
that radiation, capable of affecting sensitive plates and of
causing fluorescence in certain substances, is emitted by tubes
in which the electric discharge is passing; and that the radia-
tion proceeds from the place where the cathode rays strike
the glass walls of the tube. The X-rays, as they were called
by their discoverer, are propagated in straight lines, and can
neither be refracted by any of the substances which refract
light, nor deviated from -their course by a magnetic field ;
they are moreover able to pass with little absorption through
many substances which are opaque to ordinary and ultra-violet
light — a property of which considerable use has been made
in surgery.
The nature of the new radiation was the subject of much
speculation. Its discoverer suggested that it might prove to
represent the long-sought-for longitudinal vibrations of the
aether ; while other writers advocated the rival claims of
aethereal vortices, infra-red light, and "sifted" cathode rays.
The hypothesis which subsequently obtained general acceptance
was first propounded by Schuster* in the month following the
publication of Kontgen's researches. It is, that the X-rays are
transverse vibrations of the aether, of exceedingly small wave-
length. A suggestion which was put forward later in the year
by E. Wiechertf and Sir George StokesJ to the effect that the
rays are pulses generated in the aether when the glass of the
discharge tube is bombarded by the cathode particles, is not
really distinct from Schuster's hypothesis ; for ordinary white
light likewise consists of pulses, as Gouy§ had shown, and the
essential feature which distinguishes the Eontgen pulses is that
the harmonic vibrations into which they can be resolved by
Fourier's analysis are of very short period.
* Nature, January 23, 1896, p. 268. Fitz Gerald independently made the same
suggestion in a letter to O. J. Lodge, printed in the Electrician xxxvii, p. 372.
t Ann. d. Phys. lix (1896), p. 321.
I Xature, September 3, 1896, p. 427 : Proc. Canib. Phil. Soc. ix (1896), p. 215 ;
Mem. Manchester Lit. & Phil. Soc. xli (1896-7).
$ Journ. de Phys. v (1886), p. 354.
2 D
402 Conduction in Solutions and Gases,
The rapidity of the vibrations explains the failure of all
attempts to refract the X-rays. For in the formula
«• = !- "**
of the Maxwell-Sellmeier theory,* n denotes the frequency, and
so is in this case extremely large ; whence we have
/*' = !,
i.e., the refractive index of all substances for the X-rays is unity.
In fact, the vibrations alternate too rapidly to have an effect
on the sluggish systems which are concerned in refraction.
Some years afterwards H. Haga and C. H. Wind,f having
measured the diffraction-patterns produced by X-rays, concluded
that the wave-length of the vibrations concerned was of the
o
order of one Angstrom unit, that is about 1/6000 of the wave-
length of the yellow light of sodium.
One of the most important properties of X-rays was
discovered, shortly after the rays themselves had become known,
by J. J. Thomson,]: who announced that when they pass through
any substance, whether solid, liquid, or gaseous, they render it
conducting. This he attributed, in accordance with the ionic
theory of conduction, to " a kind of electrolysis, the molecule of
the non-conductor being split up, or nearly split up, by the
Kb'ntgen rays."
The conductivity produced in gases by this means was at
once investigated! more closely. It was found that a gas which
had acquired conducting power by exposure to X-rays lost this
quality when forced through a plug of glass-wool; whence
it was inferred that the structure in virtue of which the
gas conducts is of so coarse a character that it is unable to
survive the passage through the fine pores of the plug. The
* Cf. p. 293.
t Proceedings of the Amsterdam Acad., March 25th, 1899 (English edition, i,
p. 420), and September 27th, 1902 (English edition, v, p. 247).
% Nature, February 27, 1896, p. 391.
§ J. J. Thomson and E. Rutherford, Phil. Mag. xlii (1896), p. 392.
from Faraday to J. J. Thomson. 403
conductivity was also found to be destroyed when an electric
current was passed through the gas — a phenomenon for which
a parallel may be found in electrolysis. For if the ions were
removed from an electrolytic solution by the passage of a
current, the solution would cease to conduct as soon as
sufficient electricity had passed to remove them all ; and it
may be supposed that the conducting agents which are produced
in a gas by exposure to X-rays are likewise abstracted from it
when they are employed to transport charges.
The same idea may be applied to explain another property
of gases exposed to X-rays. The strength of the current
through the gas depends both on the intensity of the radiation
and also on the electromotive force ; but if the former factor be
constant, and the electromotive force be increased, the current
does not increase indefinitely, but tends to attain a certain
" saturation " value. The existence of this saturation value is
evidently due to the inability of the electromotive force to do
more than to remove the ions as fast as they are produced by
the rays.
Meanwhile other evidence was accumulating to show that
the conductivity produced in gases by X-rays is of the same
nature as the conductivity of the gases from flames and from
the path of a discharge, to which the theory of Giese and
Schuster had already been applied. One proof of this identity
was supplied by observations of the condensation of water-
vapour into clouds. It had been noticed long before by
John Aitken* that gases rising from flames cause precipita-
tion of the aqueous vapour from a saturated gas; and
E. von Helmholtzf had found that gases through which an
electric discharge has been passed possess the same property.
It was now shown by C. T. E. Wilson, % working in the
Cavendish Laboratory at Cambridge, that the same is true of
gases which have been exposed to X-rays. The explanation
* Trans. R. S. Edinb. xxx (1880), p. 337.
t Ann. d. Phys. xxxii (1887), p. 1.
% Proc. Roy. Soc., March 19, 1896 ; Phil. Trans., 1897, p. 265.
2 T> 2
404 Conduction in Solutions and Gases,
furnished by the ionic theory is that in all three cases the gas
contains ions which act as centres of condensation for the
vapour.
During the year which followed their discovery, the X-rays
were so thoroughly examined that at the end of that period
they were almost better understood than the cathode rays
from which they derived their origin. But the obscurity in
which this subject had been so long involved was now to be
dispelled.
Lecturing at the Eoyal Institution on April 30th, 1897,
J. J. Thomson advanced a new suggestion to reconcile the
molecular- torrent hypothesis with Lenard's observations of the
passage of cathode rays through material bodies. " We see
from Lenard's table," he said, " that a cathode ray can travel
through air at atmospheric pressure a distance of about half a
centimetre before the brightness of the phosphorescence falls to
about half its original value. Now the mean free path of
the molecule of air at this pressure is about 10~5 cm., and if a
molecule of air were projected it would lose half its momentum
in a space comparable with the mean free path. Even if we
suppose that it is not the same molecule that is carried, the
effect of the obliquity of the collisions would reduce the
momentum to half in a short multiple of that path.
" Thus, from Lenard's experiments on the absorption of the
rays outside the tube, it follows on the hypothesis that the
cathode rays are charged particles moving with high velocities
that the size of the carriers must be small compared with the
dimensions of ordinary atoms or molecules.* The assumption
of a state of matter more finely subdivided than the atom of
an element is a somewhat startling one ; but a hypothesis that
would involve somewhat similar consequences — viz. that the
so-called elements are compounds of some primordial element
— has been put forward from time to time by various
chemists."
* A similar suggestion was made by E. Wiechert, Verhandl. d. physik.-ocon.
Gesellscb. in Konigsberg, Jan. 1897.
from Faraday tcr J. J . Thomson. 405
Thomson's lecture drew from Fitz Gerald* the suggestion
that " we are dealing with free electrons in these cathode rays "
— a remark the point of which will become more evident when
we come to consider the direction in which the Maxwellian
theory was being developed at this time.
Shortly afterwards Thomson himself published an accountf of
experiments in which the only outstanding objections to the
charged-particle theory were removed. The chief of these was
Hertz' failure to deflect the cathode rays by an electrostatic
field. Hertz had caused the rays to travel between parallel
plates of metal maintained at different potentials ; but Thomson
now showed that in these circumstances the rays generate
ions in the rarefied gas, which settle on the plates, and annul
the electric force in the intervening space. By carrying the
exhaustion to a much higher degree, he removed this source of
confusion, and obtained the expected deflexion of the rays.
The electrostatic and magnetic deflexions taken together
suffice to determine the ratio of the mass of a cathode particle
to the charge which it carries. For the equation of motion of
the particle is
rar = eE .+ e[v. H],
where r denotes the vector from the origin to the position of
the particle ; E and H denote the electric and magnetic forces ;
e the charge, m the mass, and v the velocity of the particle.
By observing the circumstances in which the force #E, due to
the electric field, exactly balances the force e [v . H], due to the
magnetic field, it is possible to determine v ; and it is readily
seen from the above equation that a measurement of the
deflexion in the magnetic field supplies a relation between v
and m/e ; so both v and m/e may be determined. Thomson
found the value of m/e to be independent of the nature of the
rarefied gas : its amount was 10~7 (grammes/electromagnetic units
of charge), which is only about the thousandth part of the value
of m/e for the hydrogen atom in electrolysis. If the charge
* Electrician, May 21, 1897. t Phil. Mag. xliv (1897), p. 298.
406 Conduction in Solutions and Gases,
were supposed to be of the same order of magnitude as that on
an electrolytic ion, it would be necessary to conclude that the
particle whose mass was thus measured is much smaller than
the atom, and the conjecture might be entertained that it is the
primordial unit or corpuscle of which all atoms are ultimately
composed.*
The nature of the resinously charged corpuscles which
constitute cathode rays being thus far determined, it became of
interest to inquire whether corresponding bodies existed carrying
charges of vitreous electricity — a question to which at any rate
a provisional answer was given by W. Wienf of Aachen in the
same year. More than a decade previously E. Goldstein^ had
shown that when the cathode of a discharge-tube is perforated,
radiation of a certain type passes outward through the per-
forations into the part of the tube behind the cathode. To
this radiation he had given the name canal rays. Wien now
showed that the canal rays are formed of positively charged
particles, obtaining a value of m/e immensely larger than
Thomson had obtained for the cathode rays, and indeed of
the same order of magnitude as the corresponding ratio in
electrolysis.
The disparity thus revealed between the corpuscles of
cathode rays and the positive ions of Goldstein's rays excited
great interest ; it seemed to offer a prospect of explaining the
curious differences between the relations of vitreous and of
resinous electricity to ponderable matter. These phenomena
had been studied by many previous investigators ; in particular
Schuster,§ in the Bakerian lecture of 1890, had remarked that
" if the law of impact is different between the molecules of the
gas and the positive and negative ions respectively, it follows
that the rate of diffusion of the two sets of ions will in general
be different," and had inferred from his theory of the discharge
* The value of m/e for cathode rays was determined also in the same year by
W. Kaufmaim, Ann. d. Phys. Ixi, p. 544.
t Verh;»ndl. der physik. Gesells. zu Berlin, xvi (1897), p. 165; Ann. d. Phys.
Ixv (1898), {>. 440.
J Berlin Sitzungsber., 1886, p. 691. § Proc. R.S. xlvii (1890), p. 526.
Jrom Faraday to J . J . Thomson. 407
that " the negative ions diffuse more rapidly." This inference
was confirmed in 1898 by John Zeleny,* who showed that of
the ions produced in air by exposure to X-rays, the positive
are decidedly less mobile than the negative.
The magnitude of the electric charge on the ions of gases
was not known with certainty until 1898, when a plan for
determining it was successfully executed by J. J. Thomson.f
The principles on which this celebrated investigation was based
are very ingenious. By measuring the current in a gas which
is exposed to Rontgen rays and subjected to a known electro-
motive force, it is possible to determine the value of the product
nev, where n denotes the number of ions in unit volume of the
gas, e the charge on an ion, and v the mean velocity of the
positive and negative ions under the electromotive force. As
v had been already determined ,J the experiment led to a
determination of ne ; so if n could be found, the value of e
might be deduced.
The method employed by Thomson to determine n was
founded on the discovery, to which we have already referred,
that when X-rays pass through dust-free air, saturated with
aqueous vapour, the ions act as nuclei around which the water
condenses, so that a cloud is produced by such a degree of
saturation as would ordinarily be incapable of producing con-
densation. The size of the drops was calculated from measure-
ments of the rate at which the cloud sank ; and, by comparing
this estimate with the measurement of the mass of water
deposited, the number of drops was determined, and hence the
number n of ions. The value of e consequently deduced was
found to be independent of the nature of the gas in which the
ions were produced, being approximately the same in hydrogen
as in air, and being apparently in both cases the same as for
the charge carried by the hydrogen ion in electrolysis.
Since the publication of Thomson's papers his general
conclusions regarding the magnitudes of e and m/e for gaseous
* Phil. Mag. xlvi (1898;, p. 120. t Phil. Mag. xlvi (1898), p. 528.
+ By E. Rutherford, Phil. Mag. xliv (1897), p. 422.
408 Conduction in Solutions and Gases,
ions have been abundantly confirmed. It appears certain that
electric charge exists in discrete units, vitreous and resinous,
each of magnitude 1*5 x 10~19 coulombs approximately. Each
ion, whether in an electrolytic liquid or in a gas, carries one
(or an integral number) of these charges. An electrolytic ion
also contains one or more atoms of matter; and a positive
gaseous ion has a mass of the same order of magnitude as that
of an atom of matter. But it is possible in many ways to
produce in a gas negative ions which are not attached to atoms
of matter ; for these the inertia is only about one- thousandth
of the inertia of an atom; and there is reason for believing
that even this apparent mass is in its origin purely electrical.*
The closing years of the nineteenth century saw the founda-
tion of another branch of experimental science which is closely
related to the study of conduction in gases. When Rontgen
announced his discovery of the X-rays, and described their
power of exciting phosphorescence, a number of other workers
commenced to investigate this property more completely. In
particular, Henri Becquerel resolved to examine the radiations
which are emitted by the phosphorescent double sulphate of
uranium and potassium after exposure to the sun. The result
was communicated to the French Academy on February 24th,
1896.f " Let a photographic plate," he said, " be wrapped in
two sheets of very thick black paper, such that the plate is not
affected by exposure to the sun for a day. Outside the paper
place a quantity of the phosphorescent substance, and expose
the whole to the sun for several hours. When the plate is
developed, it displays a silhouette of the phosphorescent
substance. So the latter must emit radiations which are
capable of passing through paper opaque to ordinary light, and
of reducing salts of silver."
At this time Becquerel supposed the radiation to have been
excited by the exposure of the phosphorescent substance to the
sun ; but a week later he announced^ that it persisted for an
* Cf. p. 343. f Comptes Rendus, cxxii (1890), p. 420.
I Ibid., cxxii (March 2nd, 1896), p. 501.
from Faraday to J'. J. Thomson. 409
indefinite time after the substance had been removed from the
sunlight, and after the luminosity which properly constitutes
phosphorescence had died away ; and he was thus led to con-
clude that the activity was spontaneous and permanent. It
was soon found that those salts of uranium which do not
phosphoresce — e.g., the uranous salts, — and the metal itself, all
emit the rays ; and it became evident that what Becquerel had
discovered was a radically new physical property, possessed by
the element uranium in all its chemical compounds.
Attempts were now made to trace this activity in other
substances. In 1898 it was recognized in thorium and its
compounds;* and in the same year P. Curie and Madame
Sklodowska Curie announced to the French Academy the
separation from the mineral pitchblende of two new highly
active elements, to which they gave the names of poloniumf and
radium.}: A host of workers was soon engaged in studying the
properties of the Becquerel rays. The discoverer himself had
shown§ in 1896 that these rays, like the X- and cathode rays,
impart conductivity to gases. It was found in 1899 by
Kutherfordll that the rays from uranium are not all of the same
kind, biit that at least two distinct types are present ; one of
these, to which he gave the name a-rays, is readily absorbed ;
while another, which he named /3-radiation, has a greater
penetrating power. It was then shown by Giesel, Becquerel, and
others, that part of the radiation is deflected by a magnetic field,1T
and part is not.** After this Monsieur and Madame Curieft
found that the deviable rays carry negative electric charges,
* By Schmidt, Ann. d. Phys., Ixv (1898), p. 141 ; and by Ma.iame Curie,
Comptes Rendus, cxxvi (1898), p. 1101.
t Comptes Rendus, cxxvii (1898), p. 175. % Ibid., cxxvii (1898), p. 1215.
§ Ibid., cxxii (1896), p. 559. || Phil. Mag. (5), xlvii (1899), p. 109.
H Giesel, Ann. d. Phys. Ixix (1899), p. 834 (working with polonium);
Becquerel, Comptes Rendus, cxxix (1899), p. 996 (working with radium) ;
Meyer andv. Schweidler, Phys. Zeitschr. i (1899), p. 113 (working with polonium
and radium).
** Bc-cquerel, Comptes Rendus, cxxix (1889), p. 1205); cx\x (1900), pp. 206,
372. Curie, ibid, exxx (1900), p. 73.
ft Comptes Rendus, cxxx (1900), p. 647.
410 Conduction in Solutions and Gases.
and Becquerel* succeeded in deviating them by an electrostatic
field. The deviable or j3- rays were thus clearly of the same
nature as cathode rays ; and when measurements of the electric
and magnetic deviations gave for the ratio m/e a value of the
order 10~7, the identity of the /3-particles with the cathode-ray
corpuscles was fully established.
The subsequent history of the new branch of physics thus
created falls outside the limits of the present work. We must
now consider the progress which was achieved in the general
theory of aether and electricity in the last decade of the
nineteenth century.
* Comptes Eendus, c>xx (1900), p. 809.
CHAPTEE XII.
THE THEORY OF AETHER AND ELECTRONS IN THE CLOSING
YEARS OF THE NINETEENTH CENTURY.
THE attempts of Maxwell* and of Hertzf to extend the theory
of the electromagnetic field to the case in which ponderable
bodies are in motion had not been altogether successful.
Neither writer had taken account of any motion of the material
particles relative to the aether entangled with them, so that in
both investigations the moving bodies were regarded simply
as homogeneous portions of the medium which fills all space,
distinguished only by special values of the electric and
magnetic constants. Such an assumption is evidently incon-
sistent with the admirable theory by which FresnelJ had
explained the optical behaviour of moving transparent bodies ;
it was therefore not surprising that writers subsequent to Hertz
should have proposed to replace his equations by others
designed to agree with Fresnel's formulae. Before discussing
these, however, it may be well to review briefly the evidence
for and against the motion of the aether in and adjacent
to moving ponderable bodies, as it appeared in the last decade
of the nineteenth century.
The phenomena of aberration had been explained by Young§
on the assumption that the aether around bodies is unaffected
by their motion. But it was shown by Stokes|| in 1845 that
this is not the only possible explanation. For suppose that
the motion of the earth communicates motion to the neighbour-
ing portions of the aether ; this may be regarded as superposed
on the vibratory motion which the aethereal particles have
* Cf p. 288. t Cf. p. 365. I Cf. p. 116. $ Cf. p. 115.
|| Phil. Mag. xxvii (1845), p. 9 ; xxviii (1846), p. 76; xxix (1846), p. 6.
-412 The Theory of Aether and Electrons in the
when transmitting light : the orientation of the wave-fronts of
the light will consequently in general be altered ; and the direc-
tion in which a heavenly body is seen, being normal to the wave-
fronts will thereby be affected. But if the aethereal motion
is irrotational, so that the elements of the aether do not
rotate, it is easily seen that the direction of propagation of the
light in space is unaffected ; the luminous disturbance is still
propagated in straight lines from the star, while the normal
to the wave-front at any point deviates from this line of
propagation by the small angle ujc, where u denotes the
component of the aethereal velocity at the point, resolved at
right angles to the line of propagation, and c denotes the
velocity of light. If it be supposed that the aether near the
earth is at rest relatively to the earth's surface, the star will
appear to be displaced towards the direction in which the
earth is moving, through an angle measured by the ratio of
the velocity of the earth to the velocity of light, multiplied by
the sine of the angle between the direction of the earth's
motion and the line joining the earth and star. This is
precisely the law of aberration.
An objection to Stokes's theory has been pointed out by
several writers, amongst others by H. A. Lorentz.* This is,
that the irrotational motion of an incompressible fluid is
completely determinate when the normal component of the
velocity at its boundary is given : so that if the aether were
supposed to have the same normal component of velocity as
the earth, it would not have the same tangential component of
velocity. It follows that no motion will in general exist which
satisfies Stokes's conditions ; and the difficulty is not solved in
any very satisfactory fashion by either of the suggestions which,
have been proposed to meet it. One of these is to suppose that
the moving earth does generate a rotational disturbance, which,
however, being radiated away with the velocity of light, does not
affect the steadier irrotational motion ; the other, which was
* Archives Neerl, xxi (1896), p. 103.
Closing Years of the Nineteenth Century. 413-
advanced by Planck,* is that the two conditions of Stokes's
theory — namely, that the motion of the aether is to be
irrotational and that at the earth's surface its velocity is to be
the same as that of the earth — may both be satisfied if the
aether is supposed to be compressible in accordance with
Boyle's law, and subject to gravity, so that round the earth it
is compressed like the atmosphere ; the velocity of light being
supposed independent of the condensation of the aether.
Lorentz,f in calling attention to the defects of Stokes's
theory, proposed to combine the ideas of Stokes and Fresnel, by
assuming that the aether near the earth is moving irrotationally
(as in Stokes's theory), but that at the surface of the earth the
aethereal velocity is not necessarily the same as that of ponder-
able matter, and that (as in Fresiiel's theory) a material body
imparts the fraction (ju2 - l}/ju2 of its own motion to the aether
within it. Fresnel's theory is a particular case of this new
theory, being derived from it by supposing the velocity -potential
to be zero.
Aberration is by no means the only astronomical phenomenon
which depends on the velocity of propagation of light ; we have
indeed seent that this velocity was originally determined by
observing the retardation of the eclipses of Jupiter's satellites.
It was remarked by Maxwell§ in 1879 that these eclipses
furnish, theoretically at least, a means of determining the
velocity of the solar system relative to the aether. For if the
distance from the eclipsed satellite to the earth be divided by
the observed retardation in time of the eclipse, the quotient
represents the velocity of propagation of light in this direction,
relative to the solar system; and this will differ from the velocity
of propagation of light relative to the aether by the component,
in this direction, of the sun's velocity relative to the aether.
By taking observations when Jupiter is in different signs of the
* Of. Lorentz, Proc. Amsterdam Acad. (English ed.), i (1899), p. 443.
t Archives Neerl. xxi (1886), p. 103 : cf. also Zittinsgsversl. Kon. Ak. Amster-
dam, 1897-98, p. 266.
J Cf. p. 22. § Proc. R. S. xxx (1880), p. 108.
414 The Theory of Aether and Electrons in the
zodiac, it should therefore be possible to determine the sun's
velocity relative to the aether, or at least that component of it
which lies in the ecliptic.
The same principles may be applied to the discussion of
other astronomical phenomena. Thus the minimum of a
variable star of the Algol type will be retarded or accelerated
by an interval of time which is found by dividing the projection
of the radius from the sun to the earth on the direction from
the sun to the Algol variable by the velocity, relative to the
solar system, of propagation of light from the variable ; and thus
the latter quantity may be deduced from observations of the
retardation.*
Another instance in which the time taken by light to cross
an orbit influences an observable quantity is afforded by the
astronomy of double stars. Savaryf long ago remarked that
when the plane of the orbit of a double star is not at right
angles to the line of sight, an inequality in the apparent motion
must be caused by the circumstance that the light from the
remoter star has the longer journey to make. Yvon VillarceauJ
showed that the effect might be represented by a constant
alteration of the elliptic elements of the orbit (which alteration
is of course beyond detection), together with a periodic
inequality, which may be completely specified by the following
statement : the apparent coordinates of one star relative to the
other have the values which in the absence of this effect they
would have at an earlier or later instant, differing from the
actual time by the amount
m, - >)iz z
m} + mz ' c'
where ml and m2 denote the masses of the stars, c the velocity
of light, and z the actual distance of the two stars from each
* The velocity of light was found from observations of Algol, by C. V. L.
Charlier, Of versigt af K. Vet.-Ak. Forhandl. xivi (1889), p. 523.
t Conn, des Temps, 1830.
J Additions a la Connaissance des Temps, 1878 : an improved deduction was
given by H. Seeliger, Sitzungsberichte d. K. Ak. zu Miinchen, xix (1889), p. 19.
Closing Years of the Nineteenth Century. 415
other at the time when the light was emitted, resolved along
the line of sight. In the existing state of double-star astronomy,
this effect would be masked by errors of observation.
Villarceau also examined the consequences of supposing
that the velocity of light depends on the velocity of the source
by which it is emitted. If, for instance, the velocity of light
from a star occulted by the moon were less than the velocity of
light reflected by the moon, then the apparent position of the
lunar disk would be more advanced in its movement than that
of the star, so that at emersion the star would first appear at
some distance outside the lunar disk, and at immersion the star
would be projected on the interior of the disk at the instant of
its disappearance. The amount by which the image of the star
could encroach on that of the disk on this account could not be
so much as 0"'71 ; encroachment to the extent of more than
1" has been observed, but is evidently to be attributed for the
most part to other causes.
Among the consequences of the finite velocity of propagation
of light which are of importance in astronomy, a leading place
must be assigned to the principle enunciated in 1842 by Christian
Doppler,* that the motion of a source of light relative to an
observer modifies the period of the disturbance which is
received by him. The phenomenon resembles the depression
of the pitch of a note when the source of sound is receding from
the observer. In either case, the period of the vibrations
perceived by the observer is (c + v) / c x the natural period,
where v denotes the velocity of separation of the source and
observer, and c denotes the velocity of propagation of the
disturbance. If, e.g., the velocity of separation is equal to
the orbital velocity of the earth, the D lines of sodium in the
spectrum of the source will be displaced towards the red, as
compared with lines derived from a terrestrial sodium flame, bs
about one-tenth of the distance between them. The application
of this principle to the determination of the relative velocity of
* Abhandl. der K. Hohm. Ges. der Wissensch. (5) ii (1842), p. 465.
416 The Theory of Aether and Electrons in the
stars in the line of sight, which has proved of great service in
astrophysical research, was suggested by Fizeau in 1848.*
Passing now from the astronomical observatory, we must
examine the information which has been gained in the physical
laboratory regarding the effect of the earth's motion on optical
phenomena. We have alreadyf referred to the investigations
by which the truth of Fresnel's formula was tested. An
experiment of a different type was suggested in 1852 by
FizeauJ who remarked that, unless the aether is carried along
by the earth, the radiation emitted by a terrestrial source should
have different intensities in different directions. It was, how-
ever, shown long afterwards by Lorentz§ that such an experiment
would not be expected on theoretical grounds to yield a positive
result ; the amount of radiant energy imparted to an absorbing
body is independent of the earth's motion. A few years later
Fizeau investigatedll another possible effect. If a beam of
polarized light is sent obliquely through a glass plate, the
azimuth of polarization is altered to an extent which depends,
amongst other things, on the refractive index of the glass.
Fizeau performed this experiment with sunlight, the light
being sent through the glass in the direction of the terrestrial
motion, and in the opposite direction ; the readings seemed to
differ in the two cases, but on account of experimental difficulties
the result was indecisive.
Some years later, the effect of the earth's motion on the
rotation of the plane of polarization of light propagated along
the axis of a quartz crystal was investigated by Mascart.^f The
result was negative, Mascart stating that the rotation could
not have been altered by more than the (l/40,000)th part when
the orientation of the apparatus was reversed from that of
* An apparatus for demonstrating the Doppler-Fizeau effect in the laboratory
was constructed by Belopolsky, Astrophys. Journal xiii (1901), p. 15.
t Of. pp. 117-120. + Ann. d. Phys. xcii (1854), p. 652.
\ Proc. Amsterdam Acad. (English edition), iv (1902) p. 678.
|| Annales de Chim. (3) Ixviii (1860), p. 129; Ann. d. Phys. cxiv (1861),
p. 554.
H Annales de 1'Ec. Norm. (2) i (1872), p. 157.
Closing Years of the Nineteenth Century. 417
the terrestrial motion to the opposite direction. This was
afterwards confirmed by Lord Kayleigh,* who found that the
alteration, if it existed, could not amount to (l/100,000)th
part.
In terrestrial methods of determining the velocity of light
the ray is made to retrace its path, so that any velocity which
the earth might possess with respect to the luminiferous medium
would affect the time of the double passage only by an amount
proportional to the square of the constant of aberration.f In
1881, however, A. A. MichelsonJ remarked that the effect,
though of the second order, should be manifested by a measur-
able difference between the times for rays describing equal
paths parallel and perpendicular respectively to the direction of
the earth's motion. He produced interference-fringes between
two pencils of light which had traversed paths perpendicular
to each other ; but when the apparatus was rotated through a
right angle, so that the difference would be reversed, the expected
displacement of the fringes could not be perceived. This result
was regarded by Michelson himself as a vindication of Stokes's
theory^ in which the aether in the neighbourhood of the
dearth is supposed to be set in motion. Lorentzj), however,
showed that the quantity to be measured had only half the
value supposed by Michelson, and suggested that the negative
result of the experiment might be explained by that combina-
tion of Fresnel's and Stokes's theories which was developed in
his own memoirIF ; since, if the velocity of the aether near the
earth were (say) half the earth's velocity, the displacement of
Michelson's fringes would be insensible.
* Phil. Mag. iv. (1902), p. 215.
t The constant of aberration is the ratio of the earth's orbital velocity to the
velocity of light ; cf. supra, p. 100.
£ Amer. Journ. Sci. xxii (1881), p. 20. His method was afterwards improved :
cf. Michelson and Morley, Amer. Journ. Sci. xxxiv (1887), p. 333; Phil. Mag.
xxiv (1887), p. 449.
§ Cf. p. 411.
|| Arch. Xeerl. xxi (1886), p. 103. On the Micbelson-Morley experiment cf.
also Hicks, Phil. Mag. iii (1902), p. 9.
U Cf. p. 413.
2 E
418 The Theory of Aether and Electrons in the
A sequel to the experiment of Michelson and Morley was
performed in 1897, when Michelson* attempted to determine
by experiment whether the relative motion of earth and aether
varies with the vertical height above the terrestrial surface.
No result, however, could be obtained to indicate that the
velocity of light depends on the distance from the centre of
the earth ; and Michelson concluded that if there were no choice
<"but between the theories of Fresnel and Stokes, it would be
necessary to adopt the latter, and to suppose that the earth's
influence on the aether exends to many thousand kilometres
above its surface. By this time, however, as will subsequently
appear, a different explanation was at hand.
Meanwhile the perplexity of the subject was increased by
experimental results which pointed in the opposite direction
to that of Michelson. In 1892 Sir Oliver Lodgef observed the
interference between the two portions of a bifurcated beam of
light, which were made to travel in opposite directions round
a closed path in the space * between two rapidly rotating steel
disks. The observations showed that the velocity of light is
not affected by the motion of adjacent matter to the extent of
(l/200)th part of the velocity of the matter. Continuing his
investigations, Lodge} strongly magnetized the moving matter
(iron in this experiment), so that the light was propagated
across a moving magnetic field ; and electrified it so that the
path of the beams lay in a moving electrostatic field ; but in
no case was the velocity of the light appreciably affected.
We must now trace the steps by which theoretical physicists
not only arrived at a solution of the apparent contradictions
furnished by experiments with moving bodies, but so extended
the domain of electrical science that it became necessary to
enlarge the boundaries of space and time to contain it.
The first memoir in which the new conceptions were
unfolded-j was published by H. A. Lorentzg in 1892. The
* Amer. Journ. Sci. (4) iii (1897), p. 475.
t Phil. Trans, clxxxiv (1893), p. 727. J Ibid., clxxxix (1897), p. 149.
§ Archives Neerl. xxv (1892), p. 363 : the theory is given in eh. iv, pp. 432
et sqq.
Closing Years of the Nineteenth Century. 419
theory of Lorentz was, like those of Weber, Kiemann, and
Clausius,* a theory of electrons ; that is to say, all electro-
dynamical phenomena were ascribed to the agency of moving
electric charges, which were supposed in a magnetic field to
experience forces proportional to their velocities, and to com-
municate these forces to the ponderable matter with which
they might be associated.t
In spite of the fact that the earlier theories of electrons
had failed to fulfil the expectations of their authors, the
assumption that all electric and magnetic phenomena are due
to the presence or motion of individual electric charges was
one to which physicists were at this time disposed to give a
favourable consideration ; for, as we have seen,* evidence of
the atomic nature of electricity was now contributed by the
study of the conduction of electricity through liquids and gases.
Moreover, the discoveries of Hertz § had shown that a molecule
which is emitting light must contain some system resembling
a Hertzian vibrator; and the essential process in a Hertzian
vibrator is the oscillation of electricity to and fro. Lorentz
himself from the outset of his career! | had supposed the inter-
action of ponderable matter with the electric field to be effected
by the agency of electric charges associated with the material
atoms.
The principal difference by which the theory now advanced
by Lorentz is distinguished from the theories of Weber,
* Cf. pp. 226, 231, 262.
+ Some writers have inclined to use the term ' electron-theory ' as if it were
specially connected with Sir Joseph Thomson's justly celebrated discovery (cf . p. 407,
supra) that all negative electrons have equal charges. But Thomson's discovery,
though undoubtedly of the greatest importance as a guide to the structure of the
universe, has hitherto exercised hut little influence on general electromagnetic
theory. The reason for this is that in theoretical investigations it is customary
to denote the changes of electrons by symbols, e\, e-z, . . . ; and the equality or
non-equality of these makes no difference to the equations. To take an illustration
from Celestial Mechanics, it would clearly make no difference in the general
equations of the planetary theory if the masses of the planets happened to be
all equal.
* Cf. chapter xi.
§ Cf. pp. 357-363.
|| Verb. d. Ak. v. Wetenschappen, Amsterdam, Deel xviii (1878).
2 E 2
420 The Theory of Aether and Electrons in the
Kiemann, and Clausing, and from Lorentz' own earlier work,
lies in the conception which is entertained of the propagation
of influence from one electron to another. In the older writ-
ings, the electrons were assumed to be capable of acting on
each other at a distance, with forces depending on their
charges, mutual distances, and velocities ; in the present
memoir, on the other hand, the electrons were supposed to
interact not directly with each other, but with the medium in
which they were embedded. To this medium were ascribed the
properties characteristic of the aether in Maxwell's theory.
The only respect in which Lorentz' medium differed from
Maxwell's was in regard to the effects of the motion of bodies.
Impressed by the success of Fresnel's beautiful theory of
the propagation of light in moving transparent substances,*
Lorentz designed his equations so as to accord with that
theory, and showed that this might be done by drawing a
distinction between matter and aether, and assuming that a
moving ponderable body cannot communicate its motion to
the aether which surrounds it, or even to the aether which
is entangled in its own particles ; so that no part of the aether
can be in motion relative to any other part. Such an aether
simply space endowed with certain dynamical properties.
The general plan of Lorentz' investigation was to reduce all
the complicated cases of electromagnetic action to one simple
and fundamental case, in which the field contains only free
aether with solitary electrons dispersed in it ; the theory which
he adopted in this fundamental case was a combination of
Clausius' theory of electricity with Maxwell's theory of the
aether.
Suppose that e (x, y, z) and e(x, y', z) are two electrons.
In the theory of Clausius,f the kinetic potential of their mutual
action is
ee'
— (xx + yy + ss' - c2) ;
so when any number of electrons are present, the part of the
*Cf. pp. 116 etxqq. t Cf. p. 262.
Closing Years of the Nineteenth Century. 421
kinetic potential which concerns any one of them — say, e — may
be written
Le = e (axx + ayy + azz - c2<£),
where a and c£ denote potential functions, defined by the
equations
• / f c r r f
£— dxdy'dz, </> = \\\p- dx'dy'dz' ;
p denoting the volume-density of electric charge, and v its
velocity, and the integration being taken over all space.
We shall now reject Clausius' assumption that electrons act
instantaneously at a distance, and replace it by the assumption
that they act on each other only through the mediation of an
aether which fills all space, and satisfies Maxwell's equations.
This modification may be effected in Clausius' theory without
difficulty ; for, as we have seen,* if the state of Maxwell's
aether at any point is defined by the electric vector d and
magnetic vector h,f these vectors may be expressed in terms
of potentials a and ^ by the equations
d = c" grad <£ - a, h = curl a ;
and the functions a and <£ may in turn be expressed in terms of
the electric charges by the equations
a - JTJ \((**)'lr\ dx'dy'dz', </> = J/J |(J5)» dxdtfdsf,
where the bars indicate that the values of (pvr)' and (p)' refer
to the instant (t - r/c). Comparing these formulae with those
given above for Clausius' potentials, we see that the only change
which it is necessary to make in Clausius' theory is that of
retarding the potentials in the way indicated by L. Lorenz.J
The electric and magnetic forces, thus defined in terms of the
* Cf. pp. 298, 299.
t We shall use the small letters d and h. in place of E and H, when MC are
concerned with Lorentz' fundamental case, in which the system consists solely of
free aether and isolated electrons.
% Cf. p. 298.
422 The Theory of Aether and Electrons in the
position and motion of the charges, satisfy the Maxwellian
equations
div d = 47rc2/o,
div h * 0,
curl d = - K,
curl h = d/c2 + 47r/ov.
The theory of Lorentz is based on these four aethereal
equations of Maxwell, together with the equation which deter-
mines the ponderomotive force on a charged particle ; this,
which we shall now derive, is the contribution furnished by
Clausius' theory.
The Lagrangian equations of motion of the electron e are
^-0
fa-
and two similar equations, where L denotes the total kinetic
otential due to all causes, electric and mechanical. The
ponderomotive force exerted on the electron by the electro-
magnetic field has for its ^-component
dx ~ dt\ dx
or
fdax . dav . daz . d<t>\ dax
e{ —— x + — - it H z — c*—} — e — '-
\dx dx* dx dxj dt
which, since
reduces to
- e I & -^
or edx + e (yhz - z
so that the force in question is
ed + e [v . h].
This was Lorentz' expression for the ponderomotive force on an
Closing Years of the Nineteenth Century. 423
electrified corpuscle of charge e moving with velocity v in a field
defined by the electric force d and magnetic force h.
In Lorentz' fundamental case, which has thus been examined,
account has been taken only of the ultimate constituents of
which the universe is supposed to be composed, namely, cor-
puscles and the aether. We must now see how to build up
from these the more complex systems which are directly
presented to our experience.
The electromagnetic field in ponderable bodies, which to our
senses appears in general to vary continuously, would present a
different aspect if we were able to discern molecular structure ;
we should then perceive the individual electrons by which the
field is produced, and the rapid fluctuations of electric and
magnetic •force between them. As it is, the values furnished
by our instruments represent averages taken over volumes
which, though they appear small to us, are large compared
with molecular dimensions.* We shall denote an average
value of this kind by a bar placed over the corresponding symbol.
Lorentz supposed that the phenomena of electrostatic charge
and of conduction-currents are due to the presence or motion of
simple electrons such as have been considered above. The part
of p arising from these is the measurable density of electrostatic
charge ; this we shall denote by pi. If w denote the velocity
of the ponderable matter, and if the velocity v of the electrons
be written w + u, then the quantity pv, so far as it arises from
electrons of this type, may be written ^ w + pu. The former of
these terms represents the convection-current, and the latter
the conduction-current.
Consider next the phenomena of dielectrics. Following
Faraday, Thomson, and Mossotti,f Lorentz supposed that each
dielectric molecule contains corpuscles charged vitreously and
also corpuscles charged resinously. These in the absence of an
* These principles had been enunciated, and to some extent developed, by
J. Willard Gibbs in 1882-3 : Amer. Journ. Sci. xxiii, pp. 262, 460, xxv, p. 107 ;
Gibbs' Scientific Papei-s, ii, pp. 182, 195, 211.
t Cf. pp. 210, 211.
424 The Theory of Aether and Electrons in the
external field are so arranged as to neutralize each other's electric
fields outside the molecule. For simplicity we may suppose
that in each molecule only one corpuscle, of charge e, is capable
of being displaced from its position ; it follows from what has
been assumed that the other corpuscles in the molecule exert
the same electrostatic action as a charge - e situated at the
original position of this corpuscle. Thus if e is displaced to an
adjacent position, the entire molecule becomes equivalent to an
electric doublet, whose moment is measured by the- product of e
and the displacement of e. The molecules in unit volume, taken
together, will in this way give rise to a (vector) electric moment
per unit volume, P, which may be compared to the (vector)
intensity of magnetization in Poisson's theory of magnetism.*
As in that theory, we may replace the doublet -distribution P
of the scalar quantity p by a volume-distribution of p, determined
by the equationf
p = - div P.
This represents the part of jo due to the dielectric molecules.
Moreover, the scalar quantity pwx has also a doublet-distri-
bution, to which the same theorem may be applied ; the average
value of the part of pwx, due to dielectric molecules, is therefore
determined by the equation
pwx = - div (W.J.?) = - wx div P - (P . V) wx,
or
/ow = - div P . w - (P . V) w.
We have now to find that part of j»u which is due to dielectric
molecules. For a single doublet of moment p we have, by
differentiation,
f JJ pM dx dy dz = dp/dt,
where the integration is taken throughout the molecule; so
that
/// PM dxdydz = (d/dt) ( FP),
where the integration is taken throughout a volume V, which
*Cf. p. 64.
t We assume all transitions gradual, so as to avoid surface-distributions.
Closing Years of the Nineteenth Century. 425
encloses a large number of molecules, but which is small com-
pared with measurable quantities; and this equation may be
written
Now, if P refers to differentiation at a fixed point of space (as
opposed to a differentiation which accompanies the moving body),
we have
(£/&)*-? + (w.V)P,
and (d/dt) V = Fdiv w;
so that
/ou = P + (w . V) P + div w . P
= P + curl [P . w] + div P . w + (P . V) w,
and therefore
pu + pw = P + curl [P . w].
This equation determines the part of f>v which arises from the
dielectric molecules.
The general equations of the aether thus become, when the
averaging process is performed,
div d = 4>!r<?pi ~ 4-Trc2 div P, div h = 0,
curl d = - h,
curl h =- (1/c2) d + 47r ,
( + P + curl [P . w] I
In order to assimilate these to the ordinary electromagnetic
equations, we must evidently write
d = E, the electric force;
(1/4-7TC2) E + P = D, the electric induction ;
h = H, the magnetic vector.
The equations then become (writing p for plt as there is no
longer any need to use the subscript),
div D = p, - curl E = H,
where div H = °' curl H = 4lrS'
S = conduction-current + convection-current + D + curl [P . w].
convection-current + conduction-current .
426 The Theory of Aether and Electrons in the
The term D in S evidently represents the displacement-
current of Maxwell ; and the term curl [P . w] will be
recognized as a modified form of the term curl [D . w], which
was first introduced into the equations by Hertz.* It will
be remembered that Hertz supposed this term to repre-
sent the generation of a magnetic force within a dielectric
which is in motion in an electric field ; and that Heaviside,f by
adducing considerations relative to the energy, showed that the
term ought to be regarded as part of the total current, and
inferred from its existence that a dielectric which moves in an
electric field is the seat of an electric current, which produces
a magnetic field in the surrounding space. The modification
introduced by Lorentz consisted in replacing D by P in the
vector-product ; this implied that the moving dielectric does
not carry along the aethereal displacement, which is represented
by the term E/4?rc2 in D, but only carries along the charges
which exist at opposite ends of the molecules of the ponderable
dielectric, and which are represented by the term P. The part
of the total current represented by the term curl [P . w] is
generally called the current of dielectric convection.
That a magnetic field is produced when an uncharged
dielectric is in motion at right angles to the lines of force of a
constant electrostatic field had been shown experimentally in
1888 by Rontgen.J His experiment consisted in rotating a
dielectric disk between the plates of a condenser ; a magnetic
field was produced, equivalent to that which would be produced
by the rotation of the " fictitious charges " on the two faces of
the dielectric, i.e., charges which bear the same relation to
the dielectric polarization that Poisson's equivalent surface-
density of magnetism§ bears to magnetic polarization. If U
denote the difference of potential between the opposite coatings
of the condenser, and * the specific inductive capacity of the
dielectric, the surf ace -density of electric charge on the coatings
* Cf. p. 366. t Cf. p. 367.
I Ann. d. Phys. xxxv (1888), p. 264 ; xl (1890), p. 93.
§ Cf. p. 64.
Closing Years of the Nineteenth Century. 427
is proportional to ± t£7, and the fictitious charge on the sur-
faces of the dielectric is proportional to + (a - 1) U. It is evident
from this that if a plane condenser is charged to a given
difference of potential, and is rotated in its own plane, the
magnetic field produced is proportional to * if (as in Kowland's
experiment*) the coatings are rotated while the dielectric
remains at rest, but is in the opposite direction, and is propor-
tional to (c - 1) if (as in Kontgen's experiment) the dielectric is
rotated while the coatings remain at rest. If the coatings and
dielectric are rotated together, the magnetic action (being the
sum of these) should be independent of f — a conclusion which
was verified later by Eichenwald.f
Hitherto we have taken no account of the possible mag-
netization of the ponderable body. This would modify the
equations in the usual manner,:}: so that they finally take the
form
div D = p, (I)
div B = 0, (II)
curl H = 47rS, (III)
-curl.E = B, (IV),
where S denotes the total current formed of the displacement -
current, the convection-current, the conduction-current, and the
current of dielectric convection. Moreover, since
S =pv + d'/47rc2,
we have
div S = div pv + (l/4;rc2) div (ad/80
= div v
*Cf. p. 339.
t Ann. d. Piiys. xi (1903), p. 421 ; xiii (1904), p. 919. Eichenwald performed
other experiments of a similar character, e.g. he observed the magnetic field due
to the changes of polarization in a dielectric which was moved in a non-
homogeneous electric field.
J It is possible to construct a purely electronic theory of magnetization, a
magnetic molecule being supposed to contain electrons in orbital revolution. It
then appears that the vector which represents the average value of h. is not H,
but B.
428 The Theory of Aether and Electrons in the
which vanishes by virtue of the principle of conservation of
div 8 = 0, (V)
electricity. Thus
or the total current is a circuital vector. Equations (I) to (Y)
are the fundamental equations of Lorentz' theory of electrons.
We have now to consider the relation by which the polari-
zation P of dielectrics is determined. If the dielectric is
moving with velocity w, the ponderomotive force on unit
electric charge moving with it is (as in all theories)*
E' = E + [w . B ]. (1)
In order to connect P with E', it is necessary to consider the
motion of the corpuscles. Let e denote the charge and m the
mass of a corpuscle, (£, *?, £) its displacement from its position of
equilibrium, k* (£, 77, £) the restitutive force which retains it in
the vicinity of this point ; then the equations of motion of the
corpuscle are
ra£ + A-2£ = eEx't
and similar equations in 17 and £. When the corpuscle is set in
motion by light of frequency n passing through the medium,
the displacements and forces will be periodic functions of nt —
say,
Substituting these values in the equations of motion, we obtain
A(Jc* - mnz) -= eE«, and therefore ? (kz - tun*) = eE'x.
Thus, if N denote the number of polarizable molecules per unit
volume, the polarization is determined by the equation
* = Ne (g, TJ, ?) = JVVE7(&2 - m?i2).
In the particular case in which the dielectric is at rest, this
equatio^ gives
= (l/47rc2)E + P = (l/47rc2)E + Ne*E/(k2 - mw\
But, as we have seen,f D bears to E the ratio ^u2/47rc2, where ^
*Cf. p. 365. tCf. p. 281.
Closing Years of the Nineteenth Century. 429
denotes the refractive index of the dielectric ; and therefore the
refractive index is determined in terms of the frequency by the
equation
- mnz).
This formula is equivalent to that which Maxwell and
Sellmeier* had derived from the elastic-solid theory. Though
superficially different, the derivations are alike in their
essential feature, which is the assumption that the molecules
of the dielectric contain systems which possess free periods
of vibration, and which respond to the oscillations of the
incident light. The formula may be derived on electro-
magnetic principles without any explicit reference to electrons ;
all that is necessary is to assume that the dielectric polarization
has a free period of vibration.f
When the luminous vibrations are very slow, so that n is
small, fjr reduces to the dielectric constant ej ; so that the.
theory of Lorentz leads to the expression
e <=
* Cf. p. 293.
t A theory of dispersion, which, so far as its physical assumptions and results
are concerned, resembles that described above, was published in the same year
(1892) by Helmholtz, Berl. Ber., 1892, p. 1093, Ann d. Phys. xlviii (1893),
pp. 389, 723. In this, as in Lorentz' theory, the incident light is supposed to
excite sympathetic vibrations in the electric doublets which exist in the molecules
of transparent bodies. Helmholtz' equations were, however, derived in a different
way from those of Lorentz, being deduced from the Principle of Least Action.
The final result is, as in Lorentz' theory, represented (when the effect of damping
is neglected) by the Maxwell-Sellmeier formula. Helmholtz' theory was developed
further by Reiff, Ann. d. Phys. Iv (1895), p. 82.
In a theory .of dispersion given by Planck, Berl. Ber., 1902, p. 470, the
damping of the oscillations is assumed to be due to the loss of energy by radiation :
so that no new constant is required in order to express it.
Lorentz, in his lectures on the Iheory of Electrons (Leipzig, 1909), p. 141,
suggested that the dissipative term in the equations of motion of dielectric electrons
might be ascribed to the destruction of the regular vibrations of tit^electrons
within a molecule by the collisions of the molecule with other molecule
Some interesting references to the ideas of Hertz on the elet.:t)magnetic
explanation of dispersion will be found in a memoir by Drude, Ann. d. Phys. (6) i
(1900), p. 437.
+ Cf. p. 283.
430 The Theory of Aether and Electrons in the
for the specific inductive capacity in terms of the number and
circumstances of the electrons.*
Eeturning now to the case in which the dielectric is sup-
posed to be in motion, the equation for the polarization may be
written
from this equation, Fresnel's formula for the velocity of light in
a moving dielectric may be deduced. For, let the axis of z be
taken parallel to the direction of motion of the dielectric, which
is supposed to be also the direction of propagation of the light ;
and, considering a plane -polarized wave, take the axis of x
parallel to the electric vector, so that the magnetic vector
must be parallel to the axis of y. Then equation (III) above
becomes
equation (IV) becomes (assuming B equal to H, as is always
the case in optics),
The equation which defines the electric induction gives
IV* (1/4**)** + P.;
and equations (1) and (2) give
4arc*Px = (ft - 1) (Ex - wHy).
Eliminating Dx, Px, and Hy, we have
-iV- + '
. A»
or, neglecting w~/c2,
~dz*~ = 7 ~W ' ~~? dtfc '*
Substituting Ex = en ^ , so that V denotes the velocity
of light in the moving dielectric with respect to the fixed aether
we have
* Cf. p. 211.
t This equation was first given as a result of the theory of electrons by Lorentz
in the last chapter of his memoir of 1892, Arch. Neeii. xxv, p. 525. It was also
given by Larmor, Phil. Trans., clxxxv (1894), p. 821.
Closing Years of the Nineteenth Century. 431
or (neglecting
C fjC - 1
y - _ + £ - ^,
P M"
which is the formula of Fresnel.* The hypothesis of Fresnel,
that a ponderable body in motion carries with it the excess of
aether which it contains as compared with space free from
matter, is thus seen to be transformed in Lorentz' theory
into the supposition that the polarized molecules of the
dielectric, like so many small condensers, increase the dielectric
constant, and that it is (so to speak) this augmentation of the
dielectric constant which travels with the moving matter. One
evident objection to Fresnel's theory, namely, that it required
the relative velocity of aether and matter to be different for
light of different colours, is thus removed ; for the theory of
Lorentz only requires that the dielectric constant should have
different values for light of different colours, and of this
a satisfactory explanation is provided by the theory of
dispersion.
The correctness of Lorentz' hypothesis, as opposed to that of {
Hertz (in which the whole of the contained aether was supposed to
be transported with the moving body), was afterwards confirmed
by various experiments. In 1901 E. Blondlotf drove a current
of air through a magnetic field, at right angles to the lines of
magnetic force. The air-current was made to pass between the
faces of a condenser, which were connected by a wire, so as to be
at the same potential. An electromotive force E' would be
produced in the air by its motion in the magnetic field ; and,
according to the theory of Hertz, this should produce an
electric induction D of amount (e/47rc2) E' (where t denotes the
specific inductive capacity of the air, which is practically
unity) ; so that, according to Hertz, the faces of the condenser
should become charged. According to Lorentz' theory, on the
other hand, the electric induction D is determined by the
equation
47rc2D = E + (e - 1) E'
* Cf. p. 117. t Comptes Rendus cxxxiii (1901), p. 778.
432 The Theory of Aether and Electrons in the
where E denotes the electric force on a charge at rest, which is
zero in the present case. Thus, according to Lorentz' theory,
the charges on the faces would have only (e - l)/e of the values
which they would have in Hertz' theory ; that is, they would be
practically zero. The result of Blondlot's experiment was in
favour of the theory of Lorentz.
An experiment of a similar character was performed in
1905 by H. A. Wilson.* In this, the space between the inner
and outer coatings of a cylindrical condenser was filled with
the dielectric ebonite. When the coatings of such a con-
denser are maintained at a definite difference of potential,
charges are induced on them ; and if the condenser be rotated
on its axis in a magnetic field whose lines of force are parallel
to the axis, these charges will be altered, owing to the
additional polarization which is produced in the dielectric
molecules by their motion in the magnetic field. As before,
the value of the additional charge according to the theory of
Lorentz is (e - l)/e times its value as calculated by the theory
of Hertz. The result of Wilson's experiments was, like that of
Blondlot's, in favour of Lorentz.
The reconciliation of the electromagnetic theory with
Fresnel's law of the propagation of light in moving bodies was
a distinct advance. But the theory of the motionless aether
was hampered by one difficulty : it was, in its original form,
incompetent to explain the negative result of the experiment
of Michelson and Morley.f The adjustment of theory to
observation in this particular was achieved by means of a
remarkable hypothesis which must now be introduced.
In the issue of " Nature" for June 16th, 1892,J Lodge
mentioned that Fitz Gerald had communicated to him a new
•
suggestion for overcoming the difficulty. This was, to suppose
that the dimensions of material bodies are slightly altered
when they are in motion relative to the aether. Five months
afterwards, this hypothesis of Fitz Gerald's was adopted by
*Phii. Trans, cciv (1905), p. 121.
t Cf. p 417. J Nature, xlvi (1892), p. 165,
Closing Years of the Nineteenth Century. 433
Lorentz, in a communication to the Amsterdam Academy;*
after which it won favour in a gradually widening circle, until
eventually it came to be generally taken as the basis of all
theoretical investigations on the motion of ponderable bodies
through the aether.
Let us first see how it explains Michelson's result.
On the supposition that the aether is motionless, one of the
two portions into which the original beam of light is divided
should accomplish its journey in a time less than the other by
ur*l/c?, where w denotes the velocity of the earth, c the velocity
of light, and I the length of each arm. This would be exactly
compensated if the arm which is pointed in the direction of the
terrestrial motion were shorter than the other by an amount
w2//2c3; as would be the case if the linear dimensions of
moving bodies were always contracted in the direction of
their motion in the ratio of (1 - w'£/2c~) to unity. This is
Fitz Gerald's hypothesis of contraction. Since for the earth the
ratio w/c is only
30 km. /sec.
300,000 km./sec.'
the fraction w^jc- is only one hundred-millionth.
Several further contributions to the theory of electrons in a
motionless aether were made in a short treatisef which was
published by Lorentz in 1895. One of these related to the
explanation of an experimental result obtained some years
previously by Th. des Coudres,J of Leipzig. Des Coudres had
observed the mutual inductance of coils in different circum-
stances of inclination of their common axis to the direction of
the earth's motion, but had been unable to detect any effect
depending on the orientation. Lorentz now showed that this
could be explained by considerations similar to those which
* Verslagen d. Kon. Ak. van Wetenschappen, 1892-3, p. 74 (November 26th
1892).
t Versuch einer Theorie Jer electrischen und optischen Erscheinungen in bewegten
Eorpern, von H. A. Lorentz ; Leiden, E. J. Brill. It was reprinted by Teubner,
of Leipzig, in 1906.
+ Ann. d. Phys. xxxviii (1889), p. 73.
2 F
434 The Theory of Aether and Electrons in the
Budde and Fitz Gerald* had advanced in a similar case ; a
conductor carrying a constant electric current and moving with
the earth would exert a force on electric charges at relative
rest in its vicinity, were it not that this force induces on the
surface of the conductor itself a compensating electrostatic
charge, whose action annuls the expected effect.
The most satisfactory method of discussing the influence of
the terrestrial motion on electrical phenomena is to transform
the fundamental equations of the aether and electrons to axes
moving with the earth. Taking the axis of x parallel to the
direction of the earth's motion, and denoting the velocity of the
earth by w, we write
x = #1 + wtt y = 2/1, z = Zi,
so that (x-i, yit Zj) denote coordinates referred to axes moving
with the earth. Lorentz completed the change of coordinates
by introducing in place of the variable t a "local time" tl}
defined by the equation
t = tl + m^/c2.
It is also necessary to introduce, in place of d and h, the electric
and magnetic forces relative to the moving axes : these aret
d1 = d + [w.h]
h1 = h + (l/c2) [d.w];
and in place of the velocity v of an electron referred to^the
original fixed axes, we must introduce its velocity Vi relative to
the moving axes, which is given by the equation
V, = V - W.
The fundamental equations of the aether and electrons,
referred to the original axes, are
div d = 47re2, curl d = - h,
div h = 0, curl h = (1/c2) d +
F = d + [v . h],
where F denotes the ponderomotive force on a particle carrying
a unit charge.
* Cf . p. 263. t Cf. pp. 365, 366.
Closing Years of the Nineteenth Century. 435
By direct transformation from the original to the new
variables it is found that, when quantities of order iv*/c* and
wv/c* are neglected, these equations take the form
divj d! = 4?rc2p, curla di = - Sh^B^,
divt H! = 0, curl, hi = (1/c2) ddj/fy
F = d! + [vlthj,
where div, d, stands for
Since these have the same form as the original equations,
it follows that when terms depending on the square of the
constant of aberration are neglected, all electrical phenomena
may be expressed with reference to axes moving with the earth
by the same equations as if the axes were at rest relative to the
aether.
In the last chapter of the Versuch Lorentz discussed those
experimental results which were as yet unexplained by the
theory of the motionless aether. That the terrestrial motion
exerts no influence on the rotation of the plane of polariza-
tion in quartz* might be explained by supposing that two
independent effects, which are both due to the earth's motion,
cancel each other; but Lorentz left the question undecided.
Five years later Larmorf criticized this investigation, and
arrived at the conclusion that there should be no first-order
effect ; but LorentzJ afterwards maintained his position against
Larmor's criticism.
Although the physical conceptions of Lorentz had from
the beginning included that of atomic electric charges, the
analytical equations had hitherto involved p, the volume-density
of electric charge; that is, they had been conformed to the
hypothesis of a continuous distribution of electricity in space.
It might hastily be supposed that in order to obtain an
* Cf. p. 416. t Larmor, Aether and Matter, 1900.
J Proc. Amsterdam Acad. (English ed.), iv (1902), p. 669.
2 F 2
436 The Theory of Aether and Electrons in the
analytical theory of electrons, nothing more would be required
than to modify the formulae by writing e (the charge of an
electron) in place of pdxdydz. That this is not the case was
shown* a few years after the publication of the Versuch.
Consider, for example, the formula for the scalar potential
at any point in the aether,
where the bar indicates that the quantity underneath it is to
have its retarded value, f
This integral, in which the integration is extended over all
elements of space, must be transformed before the integration
can be taken to extend over moving elements of charge. Let
de denote the sum of the electric charges which are accounted
for under the heading of the volume- element dx'dy'dz in
the above integral. This quantity de is not identical with
~p'dx'dy'dz'. For, to take the simplest case, suppose that it is
required to compute the value of the potential-function for the
origin at the time t, and that the charge is receding from the
origin along the axis of x with velocity u. The charge which
is to be ascribed to any position x is the charge which occupies
that position at the instant t - x/c; so that when the reckoning
is made according to intervals of space, it is necessary to
reckon within a segment (x2 - a?i) not the electricity which at
any one instant occupies that segment, but the electricity which
at the instant (t - xjc) occupies a segment (x* - x\\ where x\
denotes the point from which the electricity streams to xl in the
interval between the instants (t - xz,'c) and (t - x^/c). We have
evidently
&'i - %'\ = u (%2 - %i)/c, or xz - x\ = (x2 - #0 (1 + u/c).
For this case we should therefore have
I^ ?' dx'dy'dz' = (l + ^'dx'dy'dz'.
— Xi \ cj
* E. Wiechert, Arch. Neerl. (2) v (1900), p. 549. Cf. also A. Lienard,
L' Eclairage elect, xvi (1898), pp. 5, 53, 106.
t Cf. p. 298.
Closing Years of the Nineteenth Century. 437
In the general case, it is only necessary to replace u by the
component of velocity of the electric charge in the direction of
the radius vector from the point at which the potential is to be
computed. This component may be written v cos (v . r), where r
is measured positively from the point in question to the charge,
and v denotes the velocity of the charge. Thus
cde' = {c + v cos (v . r) } ~p dx'dy'dz',
and therefore
f de'
}cr + (r.v)'
where the integration is extended over all the charges in the
field, and the bars over the letters imply that the position of
the charge considered is that which it occupied at the instant
t - r/c. In the same way the vector- potential may be shown to
have the value
r vde'
J cr + (r . v
Meanwhile the unsettled problem of the relative motion of
earth and aether was provoking a fresh series of experimental
investigations. The most interesting of these was due to
Fitz Gerald,* who shortly before his death in February, 1901,
commenced to examine the phenomena manifested by a
charged electrical condenser, as it is carried through space in
consequence of the terrestrial motion. On the assumption
that a moving charge develops a magnetic field, there will be
associated with the condenser a magnetic force at right angles
to the lines of electric force and to the direction of the
motion: magnetic energy must therefore be stored in the
medium, when the plane of the condenser includes the direc-
tion of the drift; but when the plane of the condenser is at
right angles to the terrestrial motion, the effects of the
opposite charges neutralize each other. Fitz Gerald's original
idea was that, in order to supply the magnetic energy, there
must be a mechanical drag on the condenser at the moment of
* Fitz Gerald's Scientific Writings, p. 557.
438 The Theory of Aether and Electrons in the
charging, similar to that which would be produced if the mass
of a body at the surface of the earth were suddenly to become
greater. Moreover, it was conjectured that the condenser,
when freely suspended, would tend to move so as to assume the
longitudinal orientation, which is that of maximum kinetic
energy* : the transverse position would therefore be one of
unstable equilibrium.
For both effects a search was made by Fitz Gerald's pupil
Trouton :f in the experiments designed to observe the turning
couple, a condenser was suspended in a vertical plane by a
fine wire, and charged. If the plane of the condenser were
that of the meridian, about noon there should be no couple
tending to alter the orientation, because the drift of aether due
to the earth's motion would be at right angles to this plane;
at any other hour, a couple should act. The effect to be
detected was extremely small ; for the magnetic force due to
the motion of the charges would be of order w/c, where w
denotes the velocity of the earth ; so the magnetic energy of
the system, which depends on the square of the force, would be
of order (w/c)' ; and the couple, which depends on the derivate
of this with respect to the azimuth, would therefore be likewise
of the second order in (w/c).
No couple could be detected. As the energy of the magnetic
field must be derived from some source, there seems to be no
escape from the conclusion that the electrostatic energy of a
charged condenser is diminished by the fraction (w/c)" of its
amount when the condenser is moving with velocity w at
right angles to its lines of electrostatic force. To explain this
diminution, it is necessary to admit Fitz Gerald's hypothesis
of contraction. The negative result of the experiment may be
taken to indicate^ that the kinetic potential of the system,
when the Fitz Gerald contraction is taken into account as a
* Larmor, in Fitz Gerald's Scientific Papers, p 566.
t F. T. Trouton. Trans. Roy. Dub. Soc., April, 1902; F. T. Trouton and
H. R. Noble, Phil. Trans, ccii (1903), p. 165.
J Cf. P. Langevin, Comptes Rendus, cxl (1905), p. 1171.
Closing Years of the Nineteenth Century. 439
constraint, is independent of the orientation of the plates with
respect to the direction of the terrestrial motion.
It may be remarked that the existence of the couple, had
it been observed, would have demonstrated the possibility of
drawing on the energy of the earth's motion for purposes of
terrestrial utility.
The Fitz Gerald contraction of matter as it moves through
the aether might conceivably be supposed to affect in some
way the optical properties of the moving matter; for in-
stance, transparent substances might become doubly refracting.
Experiments designed to test this supposition were per-
formed by Lord Eayieigh in 1902,* and by D. B. Brace in
1904t ; but no double refraction comparable with the propor-
tion (w/cf of the single refraction could be detected. The
Fitz Gerald contraction of a material body cannot therefore be
of the same nature as the contraction which would be produced
in the body by pressure, but must be accompanied by such
concomitant changes in the relations of the molecules to the
aether that an isotropic substance does not lose its simply
refracting character.
By this time, indeed, the hypothesis of contraction, which
originally had no direct connexion with electric theory, had
assumed a new aspect. Lorentz, as we have seen,J had
obtained the equations of a moving electric system by
applying a transformation to the fundamental equations of
the aether. In the original form of this transformation,
quantities of higher order than the first in w/c were neglected.
But in 1900 Larmorg extended the analysis so as to include
small quantities of the second order, and thereby discovered a
remarkable connexion between the equations of transforma-
tion and the equations which represent Fitz Gerald's con-
» Phil. Mag. iv (1902), p. 678.
t Phil. Mag. vii (1904), p. 317.
+ Cf. p. 434. Cf. also Lorentz, Proc. Amsterdam Acad. (English ed.), i (1899),
p. 427.
§ Larmor, Aether and Matter, p. 173.
440 The Theory of Aether and Electrons in the
traction. After this Lorentz* went further still, and obtained
the transformation in a form which is exact to all orders of the
small quantity w/c. In this form we shall now consider it.
The fundamental equations of the aether are
div d = 4irc*p, curl d = - h,
div h = 0, curl h = d/c2 + 47r/ov.
It is desired to find a transformation from the variables
x, y, z, t, p, d, h, v, to new variables xlt y^ zly th plt d,, hi, YI, such
that the equations in terms of these new variables may take
the same form as the original equations, namely :
divi dx = 47rc2jOi, curl, di = - dh^d^,
d^ h! = 0, curlj hi = (1/c2) Bdj/9^
Evidently one particular class of such transformations is
that which corresponds to rotations of the axes of coordinates
about the origin : these may be described as the linear homo-
geneous transformations of determinant unity which transform
the expression (x2 + if + zz) into itself.
These particular transformations are, however, of little
interest, since they do not change the variable t. But in place
of them consider the more general class formed of all those
linear homogeneous transformations of determinant unity in
the variables x, y, z, ct, which transform the expression
(x- + y* + z - c't") into itself : we shall show that these trans-
formations have the property of transforming the differential
equations into themselves.
All transformations of this class may be obtained by the
combination and repetition (with interchange of letters) of one
of them, in which two of the variables— say, y and z— are
unchanged. The equations of this typical transformation may
* Proc. Amsterdam Acad. (English ed.), vi, p. 809. Lorentz' work was
completed in respect to the formulae which connect pi, vi, with p, v, by Einstein,
Ann. d. Phys., xvii (1905), p. 891. It should be added that the transformation
in question had been applied to the equation of vibratory motions many years
before by Voigt, Gott. Nach. 1887, p. 41.
Closing Years of the Nineteenth Century. 441
easily be derived by considering that the equation of the
rectangular hyperbola
x2 - (cty = 1
(in the plane of the variables x, ct) is unaltered when any pair
of conjugate diameters are taken as new axes, and a new unit
of length is taken proportional to the length of either of these
diameters. The equations of transformation are thus found to be
x = Xi cosh a + cti sinh a, y = y\y
t = ti cosh a + (x}/c) sinh a, z = z,,
where a denotes a constant. The simpler equations previously
given by Lorentz* may evidently be derived from these by
writing w/c for tanh a, and neglecting powers of w/c above the
first. By an obvious extension of the equations given by
Lorentz for the electric and magnetic forces, it is seen that the
corresponding equations in the present transformation are
= dXlt hx =
dy = dyi cosh a + chzi sinh a,
dz = dzi cosh a - chv. sinh a,
hy = hyi cosh a - (l/c)dzi sinh a,
hz = hzi cosh a + (l/c)dyi sinh a.
The connexion between p and pl may be obtained in the
following way. It is assumed that if a charge e is attached to
a particle which occupies the position (f, 77, J) at the instant ty
an equal charge will be attached to the corresponding point
(f i, 771} £\) at the corresponding instant ti in the transformed
system ; so that a charge e attached to an adjacent particle
(f + A£ TI + Arj, %+ A£) at the instant t will give rise in the
derived system to a charge e at the place
V %1 A «. Ofrl A %l
fi +^Af + ^-AT/4 ^\
at the instant
* Cf. P. 434.
442 The Theory of Aether and Electrons in the
that is to say, at the place
(f ! + Af cosh a, 77, + AT?, £ + A?)
at the instant (^ - sinh a . Af/c). Thus at the instant ^, this
charge will occupy the position
(f i + Af cosh a + sinh a . Af . 0^/e, 771 + AT; + sinh a . Af . vyi/c,
£1 + Af + sinh a . A f . vai/c).
The charges corresponding to those in the original system which
were at the instant t contained in a volume A| AT? A£ will
therefore in the derived system at the instant tl occupy a volume
cosh a + sinh a . vxjc . 0 0
AS,
sinh a .Vyjc 1 0
sinh a . vzjc 0 1
or,
(cosh a + sinh a . vxjc) A? A»j A£.
Thus if jOi denote the volume-density of electric charge in the
transformed system, we shall have
pi (cosh a + sinh a . vxjc) = p ;
this equation expresses the connexion between pi and p. We
have moreover
dx fix fix dx
~ ($_ dt dt dt
vx, sech a
= c tann a + - ,
cosh a + vXl c^sinha
and similarly
vyi
cosh a + vXl c~l sinh a
and
cosh a + vXl c l sinh a
Closi?ig Years of the Nineteenth Century. 443
When the original variables are by direct substitution replaced
by the new variables in the differential equations, the latter
take the form
div! hi = 0, cur^ ht
that is to say, the fundamental equations of the aether retain
their form unaltered, when the variables are subjected to the
transformation which has been specified.
We are now in a position to show the connexion of this
transformation with Fitz Gerald's hypothesis of contraction.
Suppose that two material particles are moving along the axis
of x with velocity w - c tanh a. From the relation
vx. sech a
vx = c tanh a + — = -
cosh a + vXi c~l sinh a '
it follows that vXl is zero for each of the particles, which implies
that they are at rest relative to the new axes. Let %i and x\
denote their coordinates with respect to this latter system ; then
the coordinates of one particle at the instant ti, referred to the
original axes, will be given by the equations
x = Xi cosh a + ctl sinh o, t — t\ cosh a + Xi c~l sinh a ;
and the coordinates of the other particle will be given by
xf - x\ cosh o f cti sinh a, tf = tl cosh a + x\ c'1 sinh a ;
so that at time t the latter particle will have the coordinate x",
where
x" = of + w (t - t')
= x\ cosh a + ctl sinh a + (x - x\) sinh2 a sech a,
which gives
x" — x = (a?'i — Xi) (1 — w'/c2)^.
This equation shows that the distance between the par-
ticles in the system of measurement furnished by the original
axes, with reference to which the particles were moving with
velocity w, bears the ratio (1 - w-/c'-)^ : 1 to their distance in the
444 The Theory of Aether and Electrons in the
system of measurement furnished by the transformed axes,
with reference to which the particles are at rest. But accord-
ing to FitzGerald's hypothesis of contraction, when a material
body is in motion relative to the aether, in a direction parallel
to the axis of x, its dimensions parallel to this direction
contract in precisely this ratio; so that the equation of the
body, in terms of the coordinates x}) y^ zly which move with
it, is unaltered. Thus the hypothesis of Fitz Gerald may be
expressed by the statement .that the equations of the figures
of ponderable bodies are covariant with respect to those trans-
formations for which the fundamental equations of the aether
are covariant.
The covariance holds with respect to all linear homogeneous
transformations in the variables (x, y, z, t), of determinant
unity, which transform the expression (x"2 + y2- + z~ - c2f) into
itself. This group comprises an infinite number of transforma-
tions ; so that there are an infinite number of sets of variables
resembling (x}) ylt c,, £,), of which any one set (xr, yr, zr, tr) can
be derived from any other set (.rs, ys, zs, ts) by a transformation
of the group ; among the sets we must of course include the
original set of coordinates (x, y, z, t). But hitherto we have
proceeded on the assumption that the original set (x, y, z, t) is
entitled to a primacy among all the other sets, since the axes
(x, y, z) have been supposed to possess the special property of
having no motion relative to the aether, and the time repre-
sented by the variable t has been understood to be a definite
physical quantity. The other sets of variables (.rr, yr, sr, tr)
have been regarded merely as symbols convenient for use in
problems relating to moving bodies, but not as corresponding
to physical entities in the same degree as (x, y, z, t). "We
must now inquire whether this view is justified.
The question amounts to asking whether absolute position
in space, or at any rate absolute fixity relative to the aether, is
something which can be brought within the bounds of human
knowledge.
It is well known that the science of dynamics, as founded
Closing Years of the Nineteenth Century. 445
on Newton's laws of motion, does not supply any criterion by
which rest may be distinguished from uniform motion ; for if
the laws of motion are applicable when the position of bodies
is referred to any particular set of axes, they will be equally
applicable when position is referred to any other set of axes
which have a uniform motion of translation relative to these.
The older theories of electrostatics, magnetism, and electro-
dynamics, which are based on the conception of action at a
distance, are concerned only with relative configurations and
motions, and are therefore useless in the search for a basis of
absolute reckoning.
But the existence of an aether, which is postulated in the
undulatory theory of light, seems at first sight to involve the
conceptions of rest and motion relative to it, and thus to afford
a means of specifying absolute position. Suppose, for instance,
that a disturbance is generated at any point in free aether;
this disturbance will spread outwards in the form of a sphere ;
and the centre of this sphere will for all subsequent time
occupy an unchanged position relative to the aether. In this
way, or in many other ways, we might hope to determine, by
electrical or optical experiments, the velocity of the earth
relative to the aether.
The failure of such experiments as had been tried led
Fitz Gerald* to suggest that the dimensions of material bodies
undergo contraction when the bodies are in motion relative
to the aether. By the transformation of Lorentz and Larmor,
as we have seen, this hypothesis came to be expressed in a new
form ; namely that the equation of the figure of the body,
referred to a frame of reference moving with it, is always the
same, but that frames of reference which are in motion relative to
each other are based on different standards of length and time.
This way of regarding the matter brings into prominence the
fundamental questions involved. Before speaking of lengths
and velocities, it is necessary to examine the nature of systems
of measurement of space and time.
* Cf . p. 432.
446 The Theory of Aether and Electrons in the
Of the events with which Natural Philosophy is concerned,
each is perceived to happen at some definite location at some
definite moment. When a material object has been observed
to occupy a certain position at a certain instant, the same
object may again be observed at a subsequent instant ; but it
is impossible to determine whether the object is or is not in
the same position, since there is no obvious means of preserving
the identity of any location from one moment to another.
The physicist, however, finds it convenient to construct a
framework of axes in space and time for the purpose of fitting
his experiences into an orderly arrangement ; and the ques-
tion at issue is whether experience furnishes the means of
determining a framework completely and uniquely by
absolute properties, or whether the selection inevitably rests
on arbitrary choice and accidental circumstance.
In attempting to answer this question, it may first be
observed that the choice is always made so as to simplify
the description of natural phenomena as much as possible ;
thus, the variable which is to measure time is so chosen that
its increment in the interval between any two consecutive
beats of a pendulum is the same as its increment in the interval
between any other two consecutive beats. If the selection of
the four variables (x, y, zy t) is well made, it should be possible
to express the laws of nature by statements of a simple character,
e.g., that a body isolated from the influence of external agents
moves through equal intervals of space in equal intervals of
time.
Accepting, then, the principle that the framework of axes
is to be chosen so as to furnish the simplest possible expression
of the natural laws, it becomes of importance to determine
which of the natural laws are entitled, by reason of their
primary importance, to receive the greatest consideration.
Now many indications point to the probability that the
various types of forces which are observed in ponderable
bodies — forces of cohesion, of chemical union, and so forth —
are ultimately electric in their nature. Such an assumption
Closing Years of the Nineteenth Century. 447
would have the great advantage of explaining the contraction
postulated by Fitz Gerald, since it would represent the con-
traction as actually produced by the motion. But if this
assumption be correct, the theory of electricity and aether is
without doubt the fundamental theory of Natural Philosophy ;
and the framework of space and time should be chosen with
a view chiefly to the expression of electrical phenomena. This
may most naturally be done by stipulating that the wave-
fronts of disturbances generated in free aether shall, in the
system of length and time adopted, be accounted spheres whose
centres are at the origins of disturbance and whose radii are
proportional to the times elapsed since their initiation. Eeferred
to axes of (#,y,z,£) which satisfy these conditions, the fundamental
equations of the electric field assume the form which has been
taken as the basis of all our theoretical investigations.
Imagine now a distant star which is moving with a uniform
velocity w or c tanh a relative to this framework (x, y, z, t). The
theorem of transformation shows that there exists another
framework (a?,, y\,z\, t^, with respect to which the star is at rest,
and in which moreover the condition laid down regarding the
wave-surface is satisfied. This framework is peculiarly fitted
for the representation of the phenomena which happen on the
star ; whose inhabitants would therefore naturally adopt it as
their system of space and time. Beings, on the other hand, who
dwell on a body which is at rest with respect to the axes
(x, y, z, t) would prefer to use the latter system ; and from the
point of view of the universe at large, either of these systems
is as good as the other. The equations of motion of the aether
are the same with respect to both sets of coordinates, and
therefore neither can claim to possess the only property which
could confer a primacy — namely, an absolute relation to the
aether.*
To sum up, we may say that the phenomena whose study
is the object of Natural Philosophy take place each at a definite
* This was first clearly expressed by Einstein, Ann. d. Phys. xvii (1905),
p. 891.
448 The Theory of Aether a?id Electrons in the
location at a definite moment ; the whole constituting a four-
dimensional world of space and time. To construct a set of
axes of space and time is equivalent to projecting this four-
dimensional world into a three-dimensional world of space and
a one-dimensional world of time ; and this projection may be
performed in an infinite number of ways, each of which is
distinguished from the others only by characteristics merely
arbitrary and accidental.*
In order to represent natural phenomena without introducing
this contingent element, it would be necessary to abandon the
customary three-dimensional system of coordinates, and to
operate in four dimensions. Analysis of this kind has been
devised, and has been applied to the theory of the aether ;
but its development belongs to the twentieth century, and
consequently falls outside the scope of the present work.
From what has been said, it will be evident that, in the
closing years of the nineteenth century, electrical investigation
was chiefly concerned with systems in motion. The theory of
electrons was, however, applied with success in other directions,
and notably to the explanation of a new experimental discovery.
The last recorded observation of Faradayf was an attempt
to detect changes in the period, or in the state of polarization,
of the light emitted by a sodium flame, when the flame was
placed in a strong magnetic field. No result was obtained;
but the conviction that an effect of this nature remained to be
discovered was felt by many of his successors. TaitJ examined
the influence of a magnetic field on the selective absorption of
light ; impelled thereto, as he explained, by theoretical considera-
tions. For from the phenomenon of magnetic rotation it may be
inferred§ that rays circularly polarized in opposite senses are
propagated with different velocities in the magnetized medium ;
and therefore if only those rays are absorbed which have a
* Cf. H. Minkowski, Raum und Zeit. : Leipzig, 1909.
t Bence Jones' Life of Faraday, ii, p. 449.
J Proc. R.S. Edinb. ix (1875), p. 118.
§ Cf. pp. 174, 216.
Closing Years of the Nineteenth Century. 449'
certain definite wave-length in the medium, the period of the
ray absorbed from a beam of circularly polarized white light
will not be the same when the polarization is right-handed
as when it is left-handed. "Thus," wrote Tait, "what was
originally a single dark absorption-line might become a double
line."
The effect anticipated under different forms by Faraday and
Tait was discovered, towards the end of 1896, by P. Zeeman.*
Eepeating Faraday's procedure, he placed a sodium flame
between the poles of an electromagnet, and observed a widen-
ing of the D -lines in the spectrum when the magnetizing
current was applied.
A theoretical explanation of the phenomenon was imme-
diately furnished to Zeeman by Lorentz.f The radiation i&
supposed to be emitted by electrons which describe orbits
within the sodium atoms. If e denote the charge of an electron
of mass ra, the ponderomotive force which acts on it by virtue
of the external magnetic field is e [r . K], where K denotes the
magnetic force and r denotes the displacement of the electron
from its position of equilibrium; and therefore, if the force
which restrains the electron in its orbit be &, the equation of
motion of the electron is
mi? -t- K2r = e [f . K].
The motion of the electron may (as is shown in treatises
on dynamics) be represented by the superposition of certain
particular solutions called principal oscillations, whose distin-
guishing property is that they are periodic in the time. In order
to determine the principal oscillations, we write T^ent^- * for r,
where r0 denotes a vector which is independent of the time, and
n denotes the frequency of the principal oscillation : substitut-
ing in the equation, we have
(K- - mn*) rc = en^/^~l [r, E].
* Zittingsverslagen der Akad. v. "Wet. te Amsterdam v (1896), pp. 181, 242 ;
vi (1897), pp. 13, 99 ; Phil. Mag. (5) xliii (1897), p. 226.
t Phil. Mag. xliii (1897), p. 232.
2 G
450 The Theory of Aether and Electrons in the
This equation may be satisfied either (1) if r0 is parallel to K,
in which case it reduces to
K* - mri* = 0,
so that n has the value Km"*, or (2) if r0 is at right angles to K,
in which case by squaring both sides of the equation we obtain
the result
(V - mn*)z = #tfK\
which gives for n the approximate values KW~* ± el£/2m.
When there is no external magnetic field, so that K is zero,
the three values of n which have been obtained all reduce to
icw~V which represents the frequency of vibration of the
emitted light before the magnetic field is applied. When the
field is applied, this single frequency is replaced by the three
frequencies »cm~£, »cm"i + eK/2m, icm'i - eK/2m ; that is to say,
the single line in the spectrum is replaced by three lines close
together. The apparatus used by Zeeman in his earliest experi-
ments was not of sufficient power to exhibit this triplication
distinctly, and the effect was therefore described at first as a
widening of the spectral lines.*
We have seen above that the principal oscillation of the
electron corresponding to the frequency *cra~£ is performed in a
direction parallel to the magnetic force K. It will therefore
give rise to radiation resembling that of a Hertzian vibrator,
and the electric vector of the radiation will be parallel to the
lines of force of the external magnetic field. It follows that
when the light received in the spectroscope is that which has
been emitted in a direction at right angles to the magnetic
field, this constituent (which is represented by the middle line
of the triplet in the spectrum) will appear polarized in a plane
at right angles to the field ; but when the light received in the
spectroscope is that which has been emitted in the direction of
the magnetic force, this constituent will be absent.
We have also seen that the principal oscillations of the
electron corresponding to the frequencies Km-* ± eJ£/2m are
* Later observations, with more powerful apparatus, have shown that the
primitive spectral line is frequently replaced by more than three components.
Closing Years of the Nineteenth Century. 45 1
performed in a plane at right angles to the magnetic field K.
In order to determine the nature of these two principal oscilla-
tions, we observe that it is possible for the electron to describe
a circular orbit in 'this plane, if the radius of the orbit be
suitably chosen ; for in a circular motion the forces *2r and
.-e[r . K] would be directed towards the centre of the circle ; and
it would therefore be necessary only to adjust the radius so that
these furnish the exact amount of centripetal force required.
Such a motion, being periodic, would be a principal oscillation.
Moreover, since the force e [r . K] changes sign when the
sense of the movement in the circle is reversed, it is evident
that there are two such [circular orbits, corresponding to the
two senses in which the electron may circulate; these must,
therefore, be no other than the two principal oscillations of
frequencies K.m~^ ± el£/2m. When the light received in the
spectroscope is that which has been emitted in a direction at
right angles to the external magnetic field, the circles, are seen
edgewise, and the light appears polarized in a plane parallel to
the field ; but when the light examined is that which has been
emitted in a direction parallel to the external magnetic force,
,the radiations of frequencies Km~i ± eK/2m are seen to be
•circularly polarized in opposite senses. All these theoretical
•conclusions have been verified by observation.
It was found by Cornu* and by C. G-. W. Konigf that the
more refrangible component (i.e., the one whose period is shorter
than that of the original radiation) has its circular vibration
in the same sense as the current in the electromagnet. From
this it may be inferred that the vibration must be due to a
resinously charged electron; for let the magnetizing current
and the electron be supposed to circulate round the axis of z in
the direction in which a right-handed screw must turn in order
to progress along the positive direction of the axis of z ; then
the magnetic force is directed positively along the axis of z,
jand, in order that the force on the electron may be directed
* Comptes Rendus, cxxv (1897), p. 555.
* Ann. d. Phys. Ixii (1897), p. 240.
2G2
452 The Theory of Aether and Electrons in the
inward to the axis of z (so as to shorten the period), the charge
on the electron must be negative.
The value of e/m for this negative electron may be determined
by measurement of the separation between the components of
the triplet in a magnetic field of known strength ; for, as we
have seen, the difference of the frequencies of the outer com-
ponents is eKjm. The values of e/m thus determined agree well
with the estimations* of e/m for the corpuscles of cathode rays.
The phenomenon discovered by Zeeman is closely related to
the magnetic rotation of the plane of polarization of light. f
Both effects may be explained by supposing that the molecules
of material bodies contain electric systems which possess
natural periods of vibration, the simplest example of such a
system being an electron which is attracted to a fixed centre
with a force proportional to the distance. Zeeman's effect
represents the influence of an external magnetic field on the
free oscillations of these electric systems, while Faraday's effect
represents the influence of the external magnetic field on the
forced oscillations which the systems perform under the stimulus
of incident light. The latter phenomenon may be analysed
without difficulty on these principles, the equation of motion of
one of the electrons being taken in the form
mr + K2r = eE + e[r.H],
where m denotes the mass and e the charge of the electron,,
r its distance from the centre of force, K2r the restitutive force,
E and H the electric and magnetic forces. When the electron
performs forced oscillations under the influence of light of
frequency n, this equation becomes
(K2-m?i2)r = eE + e[r.H].
The influence of the magnetic force on the motion of the
electron is small compared with the influence of the electric
force, i.e. the second term on the right is small compared with
the first term ; so in the second term we may replace r by its
* Cf. p. 405. t Cf. pp. 213-216, 307-309, 367-370.
Closing Years of the Nineteenth Century. 453
value as found from the first term, namely, eE/(icz - run*). The
•equation thus becomes
r = K2 - mn* + (i'-wm1)1^'^'
If P denote* the electric moment" per unit volume, we have
P = ei x the number of such systems in unit volume of the
medium ;
so P must be of the form
where e evidently represents the dielectric constant of the
medium, and o- is the coefficient which measures the magnetic
rotatory power. In the magneto-optic term we may replace
H by K, the external magnetic force, since this is large com-
pared with the magnetic force of the luminous vibrations.
Thus if D denote the electric induction, we have
D = fE/47rc2 + <r [E . K].
'Combining this with the usual electromagnetic equations,
curl H = 47ri>,
curl E = - H,
we have
- curl curl E = *E/c2 + 4:r<r [E . K].
When a plane wave of light is propagated through the
medium in the direction of the lines of magnetic force, and
the axis of x is taken parallel to this direction, the equation
gives
(VEy
.and these equations, as we have seen,f are competent to explain
the rotation of the plane of polarization.
*Cf. p. 428. t Cf. p 215.
454 The Theory of Aether and Electrons in the
From the occurrence of the factor (KT - mw) in the denomi-
nator of the expression for the magneto-optic constant <r, it
may be inferred that the magnetic rotation will be very large
for light whose period is nearly the same as a free period of
vibration of the electrons. A large rotation is in fact observed*
when plane-polarized light, whose frequency differs but little
from the frequencies of the D-lines. is passed through sodium
vapour in a direction parallel to the lines of magnetic force.
The optical properties of metals may be explained, according
to the theory of electrons, by a slight extension of the analysis
which applies to the propagation of light in transparent sub-
stances. It is, in fact, only necessary to suppose that some of
the electrons in metals are free instead of being bound to the
molecules : a supposition which may be embodied in the equations
by assuming that an electric force E gives rise to a polarization.
P, where
E = aP + /3P + 7P ;
the term in a represents the effect of the inertia of the electrons ;•
the term in ]3 represents their ohmic drift ; and the term in y
represents the effect of the restitutive forces where these exist.
This equation is to be combined with the customary electro-
magnetic equations
curl H = E/c2 + 47rP, - curl E = H.
In discussing the propagation of light through the metal, we
may for convenience suppose that the beam is plane-polarized
* The phenomenon was first observed by D. Macaluso and 0. M. Corbino,.
Comptes Rendus, cxxvii (1898), p. 548, Rend. Lincei (5) vii (2) (1898), p. 293. The
theoretical explanation was supplied by AV. Voigt, Gott. Nach., 1898, p. 349,
Ann. d. Phys. Ixvii (1899), p. 345. Cf. also P. Zeeman, Proc. Amst. Acad.
v (1902), p. 41, and J. J. Hallo, Arch. N6erl. (2) x (1905), p. 148.
Voigt also predicted that if plane-polarized light, of period nearly tbe same as
that of the D radiation, were passed through sodium vapour in a magnetic field,
in a direction perpendicular to the lines of magnetic force, the velocity of propa-
gation would be found to depend on the orientation of the plane of polarization,
so that the sodium vapour would behave as a uniaxal crystal. This prediction was
confirmed experimentally by Voigt and Wiechert : cf . Voigt, Gott. Nach., 1898,
p. 355: Ann. d. Phys. Ixvii. (1899), p. 345. Cf. also A. Cotton, Cornpte*
Rendus, cxxviii (1899), p. 294, and J. Geest, Arch. Neerl. (2), x (1905), p. 291.
Closing Years of the Nineteenth Century. 455
and propagated parallel to the axis of 2, the electric vector being-
parallel to the axis of x. Thus the equations of motion reduce
to
— = -r -=-^- + 4n-
For Ex and P* we may substitute exponential functions of
where n denotes the frequency of the light, and /* the quasi-index
of refraction of the metal : the equations then give at once
<y _ i) (_ a?lt + pnS~^i + y) = 47TC2.
Writing v (1 - K v/ - 1) for p, so that v is inversely proportional
to the velocity of light in the medium, and « denotes the
coefficient of absorption, and equating separately the real and
imaginary parts of the equation, we obtain
4*0 (y- an*)
f?nz + (y - aw2)2
When the wave-length of the light is very large, the inertia
represented by the constant a has but little influence, and the
equations reduce to those of Maxwell's original theory* of the
propagation of light in metals. The formulae were experi-
mentally confirmed for this case by the researches of E. Hagen
and H. Kubensf with infra-red light ; a relation being thus
established between the ohmic conductivity of a metal .and
its optical properties with respect to light of great wave-
length.
When, however, the luminous vibrations are performed
more rapidly, the effect of the inertia becomes predominant; and
* Cf. p. 290.
t Berlin Sitzungsber., 1903, pp. 269, 410; Ann. d. Phys. xi (1903), p. 873 ;
Phil. Mag. vii (1904), p. 157.
456 The Theory of Aether and Electrons in 'the
If the constants of the metal are such that, for a certain range
of values of n, VZK is small, while v~ (I - K2) is negative, it is evident
that, for this range of values of n, v will be small and K large,
i.e., the properties of the metal will approach those of ideal
.silver.* Finally, for indefinitely great values of n, V~K is small
.and v2 (1 - K2) is nearly unity, so that v tends to unity and K
to zero : an approximation to these conditions is realized in
the X-rays.f
In the last years of the nineteenth century, attempts were
made to form more definite conceptions regarding the behaviour
of electrons within metals. It will be remembered that the
•original theory of electrons had been proposed by WeberJ for
the purpose of explaining the phenomena of electric currents
in metallic wires. Weber, however, made but little progress
towards an electric theory of metals ; for being concerned
chiefly with magneto-electric induction and electromagnetic
ponder omotive force, he scarcely brought the metal into the
discussion at all, except in the assumption that electrons of
opposite signs travel with equal and opposite velocities relative
to its substance. The more comprehensive scheme of his
successors half a century afterwards aimed at connecting in
a unified theory all the known electrical properties of metals,
such as the conduction of currents according to Ohm's law, the
thermo-electric effects of Seebeck, Peltier, and W. Thomson,
the gal vano- magnetic effect of Hall, and other phenomena which
will be mentioned subsequently.
The later investigators, indeed, ranged beyond the group
of purely electrical properties, and sought by aid of the theory of
electrons to explain the conduction of heat. The principal ground
on which this extension was justified was an experimental result
obtained in 1853 by G. Wiedemann and K. FranzJ who found
* Cf. p. 179.
t Models illustrating the selective reflexion and absorption of light by metallic
bodies and by gases were discussed by H. Lamb, Mem. and Proc. Manchester Lit.
.and Phil. Soc. xlii (1898), p. 1 ; Proc. Lond. Math. Soc. xxxii (1900), p.. 11 ; Trans.
'Camb. Phil. Soc. xviii (1900), p. 348.
+ Cf. p. 226. § Ann. d. Phys. Ixxxix (1853), p. 497.
Closing Years of the Nineteenth Century. 457
that at any temperature the ratio of the thermal conductivity
of a body to its ohmic conductivity is approximately the same
for all metals, and that the value of this ratio is proportional
to the absolute temperature. In fact, the conductivity of a
pure metal for heat is almost independent of the temperature;
while the electric conductivity varies in inverse proportion to
the absolute temperature, so that a pure metal as it approaches
the absolute zero of temperature tends to assume the character
of a perfect conductor. That the two conductivities are closely
related was shown to be highly probable by the experiments
of Tait^ in which pieces of the same metal were found to exhibit
variations in ohmic conductivity exactly parallel to variations
in their thermal conductivity.
The attempt to explain the electrical and thermal properties
•of metals by aid of the theory of electrons rests on the assump-
tion that conduction in metals is more or less similar to
conduction in electrolytes ; at any rate, that positive and
negative charges drift in opposite directions through the sub-
stance of the conductor under the influence of an electric
field. It was remarked in 1888 by J. J. Thomson,* who must
be regarded as the founder of the modern theory, that the
differences which are perceived between metallic and electro-
lytic conduction may be referred to special features in the two
cases, which do not affect their general resemblance. In
electrolytes the carriers are provided only by the salt, which
is dispersed throughout a large inert mass of solvent ; whereas
in metals it may be supposed that every molecule is capable
of furnishing carriers. Thomson, therefore, proposed to regard
the current in metals as a series of intermittent discharges,
caused by the rearrangement of the constituents of molecular
systems — a conception similar to that by which Grothussf had
pictured conduction in electrolytes. This view would, as he
showed, lead to a general explanation of the connexion between
thermal and electrical conductivities.
* J. J. Thomson, Applications of Dynamics to Physics and Chemistry, 1888,
p. 296. Cf. also Giese, Ann. d. Phys. xxxvii (1889), p. 576. t Cf. p. 78.
458 The Theory of Aether and Electrons in the
Most of the later writers on metallic conduction have pre-
ferred to take the hypothesis of Arrhenius* rather than that of
Grothuss as a pattern ; and have therefore supposed the
interstices between the molecules of the metal to be at all
times swarming with electric charges in rapid motion. In
1898 E. Eieckef effected an important advance by examining
the consequences of the assumption that the average velocity of
this random motion of the charges is nearly proportional to the
square root of the absolute temperature T. P. DrudeJ in 1900
replaced this by the more definite assumption that the kinetic
energy of each moving charge is equal to the average kinetic
energy of a molecule of a perfect gas at the same temperature ,
and may therefore be expressed in the form qT, where q denotes
a universal constant.
In the same year J. J. Thomson § remarked that it would
accord with the conclusions drawn from the study of ionization
in gases to suppose that the vitreous and resinous charges play
different parts in the process of conduction : the resinous
charges may be conceived of as carried by simple negative
corpuscles or electrons, such as constitute the cathode rays :
they may be supposed to move about freely in the interstices
between the atoms of the metal. The vitreous charges, on the
other hand, may be regarded as more or less fixed in attachment
to the metallic atoms. According to this view the transport of
electricity is due almost entirely to the motion of the negative
charges.
An experiment which was performed at this time by Eiecke||
lent some support to Thomson's hypothesis. A cylinder of
aluminium was inserted between two cylinders of copper in
a circuit, and a current was passed for such a time that the
amount of copper deposited in an electrolytic arrangement
* Cf. p. 384.
t G-ott. Nach., 1898, pp. 48, 137. Ann. d. Phys. Ixvi (1898), pp. 353, 545,
1199; ii. (1900), p. 835.
I Ann. d. Phys. (4) i (1900), p. 566 ; iii (1900), p. 369 ; vii (1902), p. 687.
§ Rapports pres. au Congres de Physique, Paris, 1900, iii, p. 138.
|| Phys. Zeitsch. iii (1901), p. 639.'
Closing Years of the Nineteenth Centwy. 459*
would have amounted to over a kilogramme. The weight of
each of the three cylinders, however, showed no measurable
change; from which it appeared unlikely that metallic con-
duction is accompanied by the transport of metallic ions.
The ideas of Thomson, Kiecke, and Drude were combined by
Lorentz* in an investigation which, as it is the most complete r
will here be given as the representative of all of them.
It is supposed that the atoms of the metal are fixed, and
that in the interstices between them a large number of resinous-
electrons are in rapid motion. The mutual collisions of the
electrons are disregarded, so that their collisions with the
fixed atoms alone come under consideration ; these are
regarded as analogous to collisions between moving and fixed
elastic spheres.
The flow of heat and electricity in the metal is
supposed to take place in a direction parallel to the axis of
xt so that the metal is in the same condition at all points of
any plane perpendicular to this direction ; and the flow is
supposed to be steady, so that the state of the system is
independent of the time.
Consider a slab of thickness dx and of unit area ; and suppose
that the number of electrons in this slab whose ^-components
of velocity lie between u and u + du, whose ^-components of
velocity lie between v and v + dv, and whose ^-components of
velocity lie between w and w + dw, is
/ (ut v, w, x) dx du dv dw.
One of these electrons, supposing it to escape collision,
will in the interval of time dt travel from (x, y, z) to (x + u dt,
y + vdt, z + wdt) : and its ^-component of velocity will at the
end of the interval be increased by an amount e Edtjm^ if ra and
e denote its mass and charge, and E denotes the electric force.
Suppose that the number of electrons lost to this group by
collisions in the interval dt is a dx du dv dw dt, and that the
* Amsterdam Proceedings (English edition) vii (1904-1905), pp. 438, 585, 684
460 The Theory of Aether and Electrons in the
number added to the group by collisions in the same interval is
b dx du dv dw dt. Then w^e have
f (u, v, w, x) + (b — a) dt = f (u + eE dt/m, v, w, x + u dt),
.and therefore
7JT ^\ J? r\ J?
i &fi of oT
b - a = — — + u — •
m du dx
Now, the law of distribution of velocities which Maxwell
postulated for the molecules of a perfect gas at rest is expressed
by the equation
rz
/= TT~^ a'3 Ne"*,
where N denotes the number of moving corpuscles in unit
volume, r denotes the resultant velocity of a corpuscle (so that
r2 = u* + v~ + w*), and a denotes a constant which specifies the
-average intensity of agitation, and consequently the temperature.
It is assumed that the law of distribution of velocities
among the electrons in a metal is nearly of this form; but a
term must be added in order to represent the general drifting of
the electrons parallel to the axis of x. The simplest assumption
that can be made regarding this term is that it is of the form
u x a function of r only ;
we shall, therefore, write
a
/ = NTT~* a'3 e «2 + u^ (r).
The value of ^ (r) may now be determined from the equation
eE'df df
b - a = — ~- + u —-;
m du dx
for on the left-hand side^ the Maxwellian term
would give a zero result, since b is equal to a in Maxwell's
.system ; thus b - a must depend solely on the term u-% (r) ; and
Closing Years cf the^ Nineteenth Century. 46 T
an examination of the circumstances of a collision, in the manner
of the kinetic theory of gases, shows that (b - a) must have the
form - ur^ (r)/l, where I denotes a constant which is closely
related to the mean free path of the electrons. In the terms
on the right-hand side of the equation, on the other hand,
Maxwell's term gives a result different from zero; and in
comparison with this we may neglect the terms which arise
from u\ (r). Thus we have
urv(r) leE d 8\ N --,
/ \m
or
lu --, fieNE d (N\ 2M* da) .
and thus the law of distribution of velocities is determined.
The electric current i is determined by the equation
i = e Jj'J uf (it, v, w) du dv dw,
where the integration is extended over all possible values of the
components of velocity of the electrons. The Maxwellian term
in f (u, v, w) furnishes no contribution to this integral, so we
have
i = e JJJ v? x (r) du dv dw.
When the integration is performed, this formula becomes
mu dx ' dxf
or
STT^W a . m /a2 dN da\
'~±&Nl + 2~e(N~fa'*adx)'
The coefficient of i in this equation must evidently represent
the ohmic specific resistance of the metal ; so if y denote the
specific conductivity, we have
4/r N
Let the equation be next applied to the case of two metals
A and B in contact at the . same . temperature T, forming an
462 The Theory of Aether and Electrons in the
•open circuit in which there is no conduction of heat or electricity
.(so that i and da/dx are zero). Integrating the equation
= m a2 clN
" 2eNdx
.across the junction of the metals, we have
Discontinuity of potential at junction = -^— log -— ;
or since f ma2, which represents the average kinetic energy of an
electron, is by Drude's assumption equal to q/T, where gr denotes
a universal constant, we have
2 q N
Discontinuity of potential at junction = ^ - T log ~- •
O €> J\ A
This may be interpreted as the difference of potential con-
nected with the Peltier* effect at the junction of two metals ;
the product of the difference of potential and the current
measures the evolution of heat at the junction. The Peltier
discontinuity of potential is of the order of a thousandth of a
volt, and must be distinguished from Volta's contact-difference
of potential, which is generally much larger, and which, as it
presumably depends on the relation of the metals to the medium
in which they are immersed, is beyond the scope of the present
investigation.
Eeturning to the general equations, we observe that the flux
of energy JFis parallel to the axis of x, and is given by the
equation
W = \m HI urif(u, v, iv) du dv dw,
where the integration is again extended over all possible values
of the components of velocity ; performing the integration, we
have
or, substituting for E from the equation already found,
TT_ ma2 . 4ml da
W = ^ - -r—r Naz -7- •
e 871-2 dx
* Cf. p. 264.
Closing Years of the Nineteenth Century. 463
Consider now the case in which there is conduction of heat
without conduction of electricity. The flux of energy will in this
case be given by the equation
W--***
K
where K denotes the thermal conductivity of the metal expressed
in suitable units ; or
3ma da
W = - K.-^— -j--
2q dx
If it be assumed that the conduction of heat in metals is
effected by motion of the electrons, this expression may be
compared with the preceding; thus we have
and comparing this with the formula already found for the
electric conductivity, we have
7 W '
an equation which shows that the ratio of the thermal to the
electric conductivity is of the form T x a constant which is the
same for all metals. This result accords with the law of
Wiedernann and Franz.
Moreover, the value of q is known from the kinetic theory of
gases; and the value of e has been determined by J. J. Thomson*
and his followers ; substituting these values in the formula for K/y,
a fair agreement is obtained with the values of K/y determined
experimentally.
It was remarked by J. J. Thomson that if, as is postulated
in the above theory, a metal contains a great number of free
electrons in temperature equilibrium with the atoms, the
specific heat of the metal must depend largely on the energy
required in order to raise the temperature of the electrons.
Thomson considered that the observed specific heats of metals
are smaller than is compatible with the theory, and was thus
* Cf. p. 407.
464 The Theory of Aether and Electrons in the
led to investigate* the consequences of his original hypothesis^
regarding the motion of the electrons, which differs from the
one just described in much the same way as Grothuss' theory. of
electrolysis differs from Arrhenius'. Each electron was now
supposed to be free only for a very short time, from the moment
when it is liberated by the dissociation of an atom to the moment
when it collides with, and is absorbed by, a different atom. The
atoms were conceived to be paired in doublets, one pole of each
doublet being negatively, and the other positively, electrified.
Under the influence of an external electric field the doublets
orient themselves parallel to the electric force, and the electrons
which are ejected from their negative poles give rise to a current
predominantly in this direction. The electric conductivity of
the metal may thus be calculated. In order to comprise the
conduction of heat in his theory, Thomson assumed that the
kinetic energy with which an electron leaves an atom is pro-
portional to the absolute temperature ; so that if one part of the
metal is hotter than another, the temperature will be equalized
by the interchange of corpuscles. This theory, like the other, leads
to a rational explanation of the law of Wiedemann and Franz.
The theory of electrons in metals has received support
from the study of another phenomenon. It was known to
the philosophers of the eighteenth century that the air near
an incandescent metal acquires the power of conducting elec-
tricity. "Let the end of a poker," wrote Canton,J "when
red-hot, be brought but for a moment within three or four
inches of a small electrified body, and its electrical power will
be almost, if not entirely, destroyed."
The subject continued to attract attention at intervals § ;
* J. J. Thomson, The Corpuscular Theory of Matter ; London, 1907.
f Cf. p. 457. \ Phil. Trans, lii (1762), p. 457.
§ Cf. E. Becquerel, Annales de Chimie xxxix (1853), p. 355 ; Guthrie, Phil.
Mag. xlvi (1873), p. 254; also various memoirs by Elster and Geitel in the
A'nnaleri d. Phys. from 1882 onwards. The phenomenon is very noticeable, as
Edison showed (Engineering, December 12, 1884, p. 553), when a filament of
carbon is hearted to incandescence in a rarefied gas. In recent years it has been
found that ions are emitted when magnesia, or any of the oxides of the alkaline
earth metals, is heated to a dull red heal. ; ,
Closing Years of the Nineteenth Century. 465
and as the process of conduction in gases came to be better
understood, the conductivity produced in the neighbourhood of
incandescent metals was attributed to the emission of electrically
charged particles by the metals. But it was not until the develop-
ment of J. J. Thomson's theory of ionization in gases that notable
advances were made. In 1899, Thomson* determined the ratio
of the charge to the mass of the resinously charged ions emitted
by a hot filament of carbon in rarefied hydrogen, by observing
their deflexion in a magnetic field. The value obtained for
the ratio was nearly the same as that which he had found for
the corpuscles of cathode rays ; whence he concluded that
the negative ions emitted by the hot carbon were negative
electrons.
The corresponding investigation-)- for the positive leak from
hot bodies yielded the information that the mass of the positive
ions is of the same order of magnitude as the mass of material
atoms. There are reasons for believing that these ions are
produced from gas which has been absorbed by the superficial
layer of the metal.J
If, when a hot metal is emitting ions in a rarefied gas, an
electromotive force be established between the metal and a
neighbouring electrode, either the positive or the negative ions
are urged towards the electrode by the electric field, and a current
is thus transmitted through the intervening space. When the
metal is at a higher potential than the electrode, the current is
carried by the vitreously charged ions : when the electrode is
at the higher potential, by those with resinous charges. In
either case, it is found that when the electromotive force is
increased indefinitely, the current does not increase indefinitely
likewise, but acquires a certain " saturation " value. The
obvious explanation of this is that the supply of ions available
for carrying the current is limited.
* Phil. Mag. xlviii (1899), p. 547.
t J. J. Thomson, Proc. Camb. Phil. Soc. xv (1909), p. 64 ; 0. W. Richardson,
Phil. Mag. xvi. (1908), p. 740.
+ Cf. Richardson, Phil. Trans, ccvii (1906), p. 1.
2 H
466 The Theory of Aether and Electrons in the
When the temperature of the metal is high, the ions
emitted are mainly negative; and it is found* that in these
circumstances, when the surrounding gas is rarefied, the satura-
tion-current is almost independent of the nature of the gas or
of its pressure. The leak of resinous electricity from a metallic
surface in a rarefied gas must therefore depend only on the
temperature and on the nature of the metal ; and it was shown
by 0. W. Richardsonf that the dependence on the temperature
may be expressed by an equation of the form
b
where i denotes the saturation-current per unit area of
surface (which is proportional to the number of ions emitted in
unit time), T denotes the absolute temperature, and A and b
are constants.!
In order to account for these phenomena, Eichardson§
adopted the hypothesis which had previously been proposed ||
for the explanation of metallic conductivity ; namely, that
a metal is to be regarded as a sponge- like structure of
comparatively large fixed positive ions and molecules, in the
interstices of which negative electrons are in rapid motion.
Since the electrons do not all escape freely at the surface, he
postulated a superficial discontinuity of potential, sufficient to
restrain most of them. Thus, let N denote the number of free
electrons in unit volume of the metal ; then in a parallelepiped
whose height measured at right angles to the surface is dx,
and whose base is of unit area, the number of electrons whose
* Cf. J. A. McClelland, Proe. Camb. Phil. Soc. x (1899), p. 241; xi (1901),
p. 296. On the results obtained when the gas is hydrogen, cf. H. A. Wilson,
Phil. Trans, ccii (1903), p. 243; ccviii (1908), p. 247; and 0. W. Richardson,
Phil. Trans, ccvii (1906), p. 1.
fProc. Camb. Phil. Soc. xi (1902), p. 286; Phil. Trans, cci (1903), p. 497.
Cf. also H. A. Wilson, Phil. Trans, ccii (1903), p. 243.
J The same law applies to the emission from other bodies, e.g. heated
alkaline earths, and to the emission of positive ions — at any rate when a steady
state of emission has been reached in a gas which is at a definite pressure.
§Phil. Trans, cci (1903), p. 497.
|| Cf. pp. 457 et sqq.
Closing Years of the^ Nineteenth Century. 467
^-components of velocity are comprised between u and u + du is
^
TT * a"1 JVe «* du dx, where | ma2 = qT,
m denoting the mass of an electron, T the absolute temperature,
and q the universal constant previously introduced.
Now, an electron whose ^-component of velocity is u will
arrive at the interface within an interval dt of time, provided
that at the beginning of this interval it is within a distance u dt
of the interface. So the number of electrons whose ^-com-
ponents of velocity are comprised between u and u + du which
arrive at unit area of the interface in the interval dt is
If the work which an electron must perform in order to escape
through the surface layer be denoted by </>, the number of
electrons emitted by unit area of metal in unit time is
therefore
c **udu, or
*»»*»*
The current issuing from unit area of the hot metal is thus
20 30
JirWeae'^, or N
where t denotes the charge on an electron. This expression,
being of the form
agrees with the experimental measures ; and the comparison
furnishes the value of the superficial discontinuity of potential
which is implied in the existence of 0.*
A few years after the date of this investigation, a plan was
* This discontinuity of potential was found to be 2*45 volts for sodium, 4-1
volts for platinum, and 6-1 volts for carbon.
468 The Theory of Aether and Electrons in the
devised and successfully carried out* for determining experi-
mentally the kinetic energy possessed by the ions after
emission. The mean kinetic energy of both negative and
positive ions was found to be the same for various metals
(platinum, gold, silver, etc.), and to be directly proportional to
the absolute temperature; and the distribution of velocities
among the ions proved to be that expressed by Maxwell's law.
The ions may therefore be regarded as kinetically equivalent
to the molecules of a gas whose temperature is the same as that
of the metal.
By the investigations which have been recorded, the hypo-
thesis of atomic electric charges has been, to all appearances,
decisively established. But all the parts of the theory of
electrons do not enjoy an equal degree of security; and in
particular, it is possible that the future may bring important
changes in the conception of the aether. The hope was
formerly entertained of discovering an aether by reference to
which motion might be estimated .absolutely ; but such a hope
has been destroyed by the researches which have sprung from
Fitz Gerald's hypothesis of contraction; and in some recent
writings it is possible to recognize a tendency to replace the
classical aether by other conceptions, which, however, have
been as yet but indistinctly outlined.
In any event, the close of the nineteenth century brought to
an end a well-marked era in the history of natural philosophy ;
and this is true not only with respect to the discoveries them-
selves, but also in regard to the conditions of scientific organiza-
tion and endeavour, which in the last decades of that period
became profoundly changed. The investigators who advanced
the theories of aether and electricity, from the time of Descartes
to that of Lord Kelvin, were, with very few exceptions,
congregated within a narrow territory : from Dublin to the
western provinces of Russia, and from Stockholm to the north
of Italy, may be circumscribed by a circle of no more than six
* 0. W. Richardson and F. C. Brown, Phil. Mag. xvi (1908), pp. 353, 890 ;
F. C. Brown, Phil. Mag. xvii (1909), p. 355 ; xviii (1909), p. 649
Closing Years of the Nineteenth Century. 469
hundred miles radius. But throughout the whole of Kelvin's
long life, the domain of culture was rapidly extending : the
learning of the Germanic and Latin peoples was carried to the
furthest regions of the earth : new universities were founded,
and inquiries into the secrets of nature were instituted in
every quarter of the globe. Let this record close with the
anticipation that fellowship in the pursuit of knowledge will
increase in the nations the spirit of generous emulation and
mutual respect.
INDEX OF AUTHORS CITED.
Abraham, M., 323, 352.
Aepinus, F. U. T., 47-52, 55.
Airy, Sir G. B., 120, 191, 214, 215.
Aitken, J., 403.
Ampere, A. M., 87-92, 312.
Ango, P., 24.
Arago, F., 86, 114, 116, 121, 122, 136,
173.
Arrhenius, S., 383, 384.
Aschkinass, E., 295.
Aubel, E. van, 322.
Anlinger, E., 356.
Bacon, Sir F., Lord Verulam, 2, 3, 33.
Banks, Sir J., 75.
Bartholin, E., 25.
Bartoli, A., 306.
Basset, A. B., 370.
liatelli, A., 267.
Becearia, G. B., 49, 53, 67, 75.
Becher, J. J., 36.
Becquerel, A. C., 93, 94.
Becquerel, E., 464.
Bec4uerel, H., 408, 409, 410.
Belopolsky, A., 416.
Bennet, A., 73, 304.
Bernoulli, D., 9, 50.
Bernoulli, John (the elder), 101..
Bernoulli, John (the younger), 9, 100-
102.
Berthollet, A., 112.
Berzelius, J. J., 80-83.
Betti, E., 65.
Bezold, W. v., 357.
Biot, J. B., 86, 114, 174.
Bjerknes, C. A., 316, 317.
Bjeiknes, V., 303.
Blondlot, R., 431, 432.
Boerhaave, H., 35.
Boltzmann, L., 206, 322, 325, 356.
Boscovich, R. G., 33, 161, 217.
Bottomley, J. T., 297.
Boussinesq, J., 185-187, 215.
Boyle, U., 11, 17, 31-33, 35.
Brace, D. B., 439.
Bradley, J., 99, 100.
Brewster, Sir D., Ill, 113, 134, 177.
Brougham, H., Lord, 108.
Brouncker, Viscount, 10.
Brown, F. C., 467.
Brugmans, A., 56, 218.
Budde, E., 263.
Buffon, G. L. L., Comte de, 48.
Cabeo, N., 31, 189.
Campbell, L., 283, 296.
Canton, J., 50, 464.
Carlisle, Sir A., 75, 78.
Cascariolo, V., 19, 20.
Cassini, G. D., 22.
Cauchy, A. LM 132, 139, 142-150, 158,
159, 161, 163, 165, 167, 170, 177-
179, 182, 183, 294.
Cavendish, Lord C., 51.
Cavendish, Hon. H., 51-54, 75. 94,
167, 207.
Chandler, S. C., 100.
Charlier, C. V. L., 190, 269.
Chasles, M. 190, 269.
Chattock, A. P., 357.
Chladni, E. F. F., 110.
Christiansen, C., 291.
Christie, S. H., 213.
Clausius, R., 231, 234, 261-263, 274,
373, 420-422.
ColUnson, P., 43, 46.
Corbino, 0. M., 454.
Cornu, M. A., 216, 282, 451.
Cotton, A., 454.
Coulomb, C. A., 56-59.
Courtivron, G., Marquis de, 104.
Crookes, Sir W., 306, 394, 395,
472
Index.
Cruickshank, W., 75, 76.
Gumming, J., 93, 266.
Cunaeus, 41.
Curie, P., 235, 409.
Curie, Mme. S., 409.
Daniell, F., 206, 373.
Darbishire, F. V., 204.
Davy, J., 194.
Davy, SirH., 76-78, 80, 94, 95, 188,
197, 372, 392.
De La Hire, nee La Hire.
Delambre, J. B. J., 22.
De la Eive, A., 79, 80.
De la Rive, L., 197, 201, 202, 360.
Desaguliers, J. T., 37-39.
Descartes, R., 2-9, 38, 85.
Des Coudres, T., 433.
Desormes, C. B., 84.
Digby, K., 31.
Donati, L., 349.
Doppler, C., 415.
Drude, P., 370, 429, 458, 459.
Du Fay, C. F., 39, 40, 44, 303.
Duhem, P., 281.
Dulong, P. L., 132.
Ebert, 396, 399.
Edison, T., 464.
Eichenwald, A., 339, 427.
Einstein, A., 440, 447.
Elster, J., 464.
Ettingshausen, A. v., 322.
Euler, L., 9, 66, 103, 104, 304.
Ewing, J. A., 237.
Fabroni, G., 71, 76.
Faraday, M., 45, 58, 82, 85, 188-221,
244, 248, 254, 264, 269, 271, 272,
275, 276, 279, 284, 286, 288, 300,
339, 349, 350, 373, 391, 448.
Fechner, G. T., 98, 201, 225, 226.
Fermat, P. de, 9, 10, 102, 103.
Fitz Gerald, G. F., 157, '263, 308, 318,
319, 323, 324, 325, 327, 332, 333,
334, 340, 341, 345-347, 361, 364,
367, 368, 370, 396, 401, 405, 432,
433, 437, 438.
Fi/eau, H. L., 117, 136, 254, 282, 283,
416.
Fiippl, A., 264.
Foucault, L., 136, 282, 283.
Fourcroy, A. F. de, 93.
Fourier, J., Baron, 95, 132, 139, 256.
Franklin, B., 41-51,84, 103.
Franklin, W. S., 264.
Franz, R., 456, 457.
Fresnel, A., 24, 28, 113-136, 148, 174.
Frohlich, I., 263.
Galileo, G., 21.
Galitzine, B., 306.
Gallop, E. G., 237, 238.
Galvani, L., 67-71.
Garnett, W., 283, 296.
Gauss, K. F., 58S 225-231, 268, 269.
Gautherot, N., 94.
Guy-Lussac, L. J., 199.
Geest, J., 454.
Geissler, H., 392.
Geitel, H., 464.
Gibbs, J. Willard, 283, 297, 378, 380,
423.
Giese, W., 397, 398, 457.
Giesel, F., 409.
Gilberd or Gilbert, W., 8, 29-31.
Glasenapp, S. von, 22.
Glazebrook, R. T., 131, 160, 164, 172,
173, 370.
Goldhammer, D. A., 370, 371.
Goldstein, E., 393, 396, 406.
Gounelle, E., 254.
Gouy, G., 401.
Grassnmnn, H., 91, 231.
Gray, S., 37, 38, 49.
s'Gravesande, W. J., 32, 34, 36, 108.
Green, G., 64-66, 150-154, 158, 161-
165, 167, 168, 170, 179, 296.
Gren, F. A. C., 70, 74.
Grimaldi, F. M., 11.
Grothuss, T., Freiherr v., 78-81.
Grove, Sir W. R., 241.
Guericke, 0. v., 37-
Guthrie, F., 464.
Hachette, J. N. P., 84.
Haga, H., 402.
Hagen, E., 455.
Hall, E. H., 320-323.
Halley, E., 99, 106.
Hallo, J. J., 454.
Index.
473
Hallwachs, W., 399, 400.
Hamilton, Sir W. R., 131, 139.
Hansteen, C., 84.
Hasenohrl, F., 370.
Hastings, C. S., 131, 172.
Hattendorf, K., 231.
Hauksbee, F., 39, 390.
Heaviside, 0., 341-344, 366, 367.
Heliodorus of Larissa, 10.
Helmholtz, H. v., 196, 205, 229, 240-
243, 247, 253, 261, 274, 275, 288,
293, 297, 307, 312, 325, 337-339,
353, 357, 378-382, 386, 397, 429.
Helmholtz, R. v., 403.
Henry, J., 193, 253, 358.
Hero of Alexandria, 10.
Herschel, Sir J., 174, 213.
Herschel, Sir W., 54.
Hertz, H., 347, 353-366, 396, 399, 405,
411, 429, 431, 432.
Hicks, W. M., 316, 327, 328, 333-336,
417.
Hittorf, W., 374, 375, 393, 396, 398,
399.
Hoek, M., 118, 120.
Holzmiiller, G., 233.
Homberg, W., 34, 35, 303.
Hooke, R., 11-17, 33, 36, 122.
Hopkinson, J., 321.
Horsley, S., 17.
Howard, J. L., 363.
Hughes, D. E., 237.
Hull, G. F., 307.
Hutchinson, C. T., 339.
Huygens, C., 6, 17, 22-28, 99, 145,
181.
Jacobi, M. H., 201.
Jaequier, F., 54.
Jenkin, W., 193, 194.
Joule, J. P., 239, 240, 242.
Kahlbaum, G. W. A., 204.
Kaufmann, W., 343, 406.
Kelvin, see Thomson, W.
Kepler, J., 304.
Kerr, J., 338, 368, 370.
Kirchhoff, G., 250-252, 257-259, 260-
261, 312.
Kleist, E. G. v., 41.
Koenigsberger, L., 241.
Kohlrausch, F. W., 374.
Kohlrausch, R., 251, 252, 259.
Kolacek, F., 323.
Konig, C. G. W., 451.
Korn, A, 317.
Korteweg, D. J., 91.
Kundt, A., 291.
Kiistner, F., 100.
Lagrange, J. L., 60, 103, 139.
La Hire, P. de, 22, 189.
Lamb, H., 261, 344, 456.
Lambert, J. H., 55.
Langevin, P., 438.
Laplace, P. S., Marquis de, 60, 61, 109,
110, 112, 114, 132, 139, 232, 233.
Larmor, Sir J., 118, 167, 319, 323, 343,
362, 363, 368, 370, 430, 435, 438,
439.
Lavoisier, A. L., 33, 35, 36.
Leahy, A. H., 317, 318.
Leathern, J. G., 370.
Lebedew, P., 307.
Lecher, E., 360,
Lee, A., 361.
Legendre, A.M., 60.
Lenard, P., 396, 404.
Lenz, E., 222.
Leroux, F. P., 291.
Le Seur, T., 54.
Le Verrier, U. J. J., 234.
Levy, M., 234.
Lienard, H., 436.
Lippmann, G., 375-378.
Lloyd, H., 131.
Lodge, Sir 0. J., 311, 320, 357, 358,
363, 401, 418, 432.
Lorberg, H., 231, 356.
Lorentz, H. A., 290, 322, 337, 412,
413, 416-449, 459-463.
Lorenz, L., 169, 297-300, 324, 361.
Macaluso, D., 454.
Macaulay, Lord, 108.
McClelland, J. A., 466.
MacCullagh, J., 130, 148-150, 154-157,
175-179, 289, 295, 296.
Macdonald, H. M., 348.
2 I
474
Index.
Mairan, J. J. de, 303.
Malus, E. L., Ill, 112, 177.
Marcet, M., 188.
Marianini, S., 201.
Mascart, E., 121, 416.
Maupertuis, P. L. M. de, 102, 103.
Maxwell, J. Clerk, 52, 65, 92, 102, 167,
190, 215, 237, 250, 263, 268-313,
321, 333, 337, 348, 365, 397, 411,
413, 460.
Mayer, E., 242.
Mayer, T., 55.
Melvill, T., 104.
Meyer, S., 409.
Michel), J., 54, 55, 116, 161, 167, 217,
303.
Michelson, A. A., 117, 283, 417, 418.
Miller, W. A., 373.
Minkowski, H., 448.
Morichini, D. P., 213.
Morley, E. W., 117, 417,418.
Morton, W. B., 343.
Moser, J., 381.
Mossotti, F. 0., 211, 286.
Mottelay, P. F., 8.
Musschenbroek, P. van, 41, 55.
Navier, C. L. M. H., 138-140.
Nernst, W., 380, 386-389.
Neumann, C., 176, 215, 216, 312.
Neumann, F. E., 143, 148, 149, 184,
222--22S, 261.
Newcomb, S., 283.
Newton, Sir I., 9, 15-21, 28, 31-34,
53, 106, 107.
Nichols, E. F., 307.
Nichols, E. L., 264.
Nicholson, W., 75, 77, 78.
Niven, C., 344.
Nobili, L., 193.
Noble, H. E., 438.
Nollet, J. A., 40-42, 47, 48, 391.
Nyren, M., 100.
O'Brien, M., 142, 184.
Oersted, H. C., 84-87.
Ohm, G. S., 95-98, 201,.
Oppenheim, S., 234.
Ostwald, W., 384.
Palmaer, W., 381.
Pardies, I. G., 24.
Peacock, G., 108, 125.
Pearson, K., 140, 164, 185, 361.
Peltier, J. C., 264-267.
Pender, H., 339.
Peregrinus, P., 7, 8, 189.
Pen-in, J., 400.
Perrot, A., 397.
Pfaff, C. H., 76, 201.
Planck, M., 378, 386, 413, 429.
Pliicker, J., 219, 220, 392, 393.
Poggendorff, J. C., 201.
Poincare, H., 352, 360, 361.
Poisson, S. D., 59-65, 114, 115, 134r
139-141, 245, 246.
Pouillet, C. S. M., 193, 373.
Poynting, J. H., 347-350.
Preston, S. T., 193.
Priestley, J., 36, 50-54, 75, 161, 283,
303, 304, 393.
Eankine, W. J. M., 140, 171.
Eaoult, F., 383.
Eayleigh, J. W. Strutt, Lord, 167, 170r
171, 179, 181, 283, 290, 292, 344,
417, 439.
Eeich, F., 219.
Eeiff, E., 319, 370, 429.
Eespighi, L., 120.
Eichardson, 0. W., 465, 466, 467.
Eiecke, E., 395, 458, 459.
Eiemann, B., 231, 234, 261-263, 268,
269, 297, 324.
Eitchie, W., 194.
Eitter, J. W., 75, 375.
Eobison, J., 51, 116.
Eoemer, 0., 22, 99.
Eoget, P. M., 78, 202, 203.
Eontgen, W. C., 400, 401, 426, 427.
Eowlaml, H. A., 321, 339, 344, 368,
369, 370, 427.
Eubens, H., 295, 455.
Eumford, B. Thompson, Count, 35, 188,
242.
Eutherford, E., 402, 407, 409.
Saint-Venant, B. de, 163, 164.
Sampson, E.A., 22.
Index.
475
Sarasin, E., 360.
Savart, F., 86.
Savary, F., 253, 414.
Scheele, K. W., 35, 36.
Schiller, N., 338.
Schmidt, G. C., 409.
Schonbein, C. F., 204.
Schuster, A., 343, 398, 399, 401, 406.
Schweidler, E. v., 409.
Searle, G. F. C., 343.
Seebeck, T. J., 92, 93, 265, 266.
Seegers, 233.
Seeliger, H., 100, 414.
Sellmeier, W., 293, 295.
Snell, W., 6.
Socin, A., 50.
Somerville, M., 213.
Sommerfeld, A., 319.
Spence, 43.
Stahl, G. E., 36.
Stefan, J., 306, 345.
Stokes, Sir G. G., 117, 131, 132, 137,
141, 167-169, 171, 172, 197, 255,
273, 291, 296, 401, 411, 412.
Stoney, G. Johnstone, 397.
Struve, W., 100.
Sulzer, J. G., 67.
Symmer, R., 56.
Tait, P. G., 91, 267, 395, 448, 449,
457.
Talcott, 100.
Taylor, B., 35.
Thenard, L. J., 93, 199.
Thomson, Sir J. J., 167, 326, 339, 340,
343, 344, 350-353, 365, 370, 396,
400, 402-407, 419, 457, 458, 459,
463, 464, 465.
Thomson, W. (Lord Kelvin), 52, 101,
140, 157-161, 165-168, 173, 174,
209, 211, 219, 240-250, 253-257,
265-267, 269, 270, 274-276, 279,
284, 286, 292, 294, 297, 310, 311,
313, 315, 316, 325, 326, 328-332,
336, 370, 400.
Tisserand, F., 233, 234.
Todhunter, I., 140.
Torricelli, E., 23.
Trouton, F. T., 364, 438.
Tyndall, J., 219.
Van Marum, M., 57, 76, 84.
Van 't Hoff, J. H., 388.
Varley, C. F., 376, 393.
Vauquelin, L. N., 93.
Verdet, E., 125, 215, 216.
Villarceau, Y., 414, 415.
Voigt, W., 370, 440, 454.
Volta, A., 57, 70-76, 195, 252, 375.
Walker, G. T., 353.
Wangerin, A., 143.
Warburg, E., 380.
Watson, Sir W., 42, 43, 48, 51, 254,
390, 391.
Watson, H. W., 288.
Weber, W., 193, 2J9, 225-236, 259,
261-263, 268, 282, 283, 356, 456.
Welby, F. A., 241.
Wheatstone, Sir C., 98, 254.
Whiston, W., 9.
Wiechert, E., 401, 404, 436, 454.
Wiedemann, E., 396, 399.
Wiedemann, G., 456, 457.
Wien, W., 343, 406.
Wiener, 0., 364.
Wilberforce, L. R., 311.
Wilcke, J. K., 48, 50, 56.
Williams, A., 37.
Williamson, A., 372, 373.
Wilson, C. T. R., 403.
Wilson, H. A., 432, 466.
Wilson, P., 116.
Wind, C. H., 370, 402.
Witte, H., 324.
Wollaston, W. H., 76, 77, 109, 252.
Young, T., -28, 105-111, 115, 121-123,
125, 132, 134, 136, 167, 304.
Zeeman, P., 449, 450, 451.
Zeleny, J., 407.
Printed by PoNSONBY & GIBBS, University Press, Dublin.
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